aa r X i v : . [ m a t h . G M ] F e b Examples of Morphological Calculus
Frank Sommen
Abstract.
In this paper we present an introduction to morphologicalcalculus in which geometrical objects play the rule of generalised naturalnumbers.
Mathematics Subject Classification (2010).
Keywords.
Lie groups; homogeneous spaces; Fibre bundles; Poincar´epolynomial; twistor space.
Contents1. Introduction 12. The Real Line 53. Carthesian Space, Spheres, Projective Spaces 84. Groups and Homogeneous Spaces 195. Nullcones and Things 296. Conclusions and Remarks 407. Outlook 448. Acknowledgement 45References 45
1. Introduction
Morphological calculus in an extension of the calculus of natural numbers1 , , • the natural numbers 1 , , , . . . • the real line R • the set of natural numbers N • carthesian spaces R , R , . . . • projective spaces RP n , CP n , . . . • Spheres S n − , C S n − • Groups like SO ( n ) , U ( n ) , GL ( n, R ) , . . . Frank Sommen • Graßmann manifold G ( m, k, R )and other groups and homogeneous spaces. In fact any kind of geometricalobjects can be added to the list.The rules for morphological calculus extend the rules for calculating withnatural numbers. We have1. The addition t + t + · · · + t k The terms t , . . . , t k are supposed to represent morphological objectsand the addition represents any object that can be formed by making adisjoint union of the objects t , . . . , t k and glueing them together whenpossible. This glueing process is itself not part of the calculus so therein no unique way to do it and one also needn’t do it; one can simplyput the objects t , . . . , t k in a list, as the language of calculus suggests.For example 1 + 2 + 3 + 4 can be visualised as a triangle of 10 points:1+(1+1)+(1+1+1)+(1+1+1+1).The terms t , . . . , t k in an addition may simply be names for morpho-logical objects but, they also could be expressions between brackets likein: 5 + (3 + 1) + 2 + (1 + 2 + 7) . The material between brackets is interpreted as a single morphologicalobject.2. The subtraction t − t This means that the object t is deleted from the object t . For example3 − R − t − t = c or such that t canbe written as c + t . There may not be a morphologically acceptablesolution for this. For example0 = 1 − − , − , . . . are no objects of morphologicalcalculus although they may be meaningful as actions: − − t − t here, but ofcourse one may also consider extended expression like 7 − − − v · w For the natural numbers, the multiplication is a notation for repeatedaddition, so for example • · a = a • · a = a + a • · a = a + a + a xamples of Morphological Calculus 3etc. In other words, the meaning of multiplication is in fact determinedby the rule of distributivity( t + t + · · · + t k ) · w = t · w + t · w + · · · + t k · w. In morphological calculus, the product v · w means that every point ofthe object v is replaced by a copy of w and then all those copies of w are possibly glued together in some way that is not specified by thelanguage of calculus.Typical examples are: the carthesian product v × w , a fibre bundle E = M · F with base space M and fibre F. One can also consider long multiplication like v · v · · · v k that maycorrespond to iterated fibre bundles. note that the fibre bundle inter-pretation is only an option; it isn’t a must and it will not always beavailable.4. The division v/w Like the subtraction, also the division is seen as a problem: to find amorphological object c for which v = c · w . Any good solution to this willbe denoted as v/w and again there may not always be a solution. Forexample rational numbers like 1 / , / , / / / / / . Hence the language of morphological calculus is similar to that of naturalnumbers. There are however some aspects of language of calculus that causedilemmas and also need more explanation1. Names, definitions, substitutions.Every morphological object has a name attached to it. For example1 , , , . . . the natural members, R the real line and so on. Then everyname is given a definition or several definitions of the formName = Expressionthe first main examples being the definitions of the natural numbers2 = 1 + 1 , , N appears somewhere in an expression E , i.e., E = E ( N )and when one has a definition N = Expr. ; then one may perform thesubstitution E ( N ) = E ((Expr . )), i.e., replacing the name N by the ex-pression ( Expr. ) between brackets. Later on one may investigate howand when brackets may be removed. We do not use brackets in a redun-dant manner like e.g. (7) is not used, (Name) is not used ((Expr . )) isnot used, Name = (Expr . ) is not used. Frank SommenExample: (The Fibonacci trees)These morphological structures are defined by f , f = 1 , f n = f n − + f n − leading to the solutions f = 1 + 1 f = (1 + 1) + 1 f = ((1 + 1) + 1) + (1 + 1) f = (((1 + 1) + 1) + (1 + 1)) + ((1 + 1) + 1) , so what appears here are not just the Fibonacci numbers 2 , , , , butthe tree-like structures that give rise to these numbers if one removes thebrackets. This tree-like structure is a typical example of a morphologicalobject.2. Commutativity, AssociativityIn morphological calculus the addition t + · · · + t k is in the first placea listing of objects; it is not commutative. Also within an addition onemay consider expressions between brackets and since brackets refer tomorphological objects one can’t just ignore them; the addition is not justassociative. On the other hand, for the natural numbers the addition alsorefers to the total quantity or sum. For example the total quantity of5 + (3 + 1) + 2 may be evaluated as:5 + (3 + 1) + 2 = (1 + 1 + 1 + 1 + 1) + ((1 + 1 + 1) + 1) + (1 + 1)= (1 + 1 + 1 + 1 + 1) + (1 + 1 + 1 + 1) + (1 + 1)= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 11 , so it requires substitutions 5 → (1 + 1 + 1 + 1 + 1) etc. and deleting thebrackets. So the total quantity is evaluated within the language of cal-culus and not in some outside theory. It corresponds to a morphologicalprocess in which the morphological structure is constantly changed tothe extent that in the final evaluation of the quantity, the identity of thenumbers 5 , , . . . as well as their place in the context is lost. Commuta-tivity, substitutions and putting and deleting brackets are guaranteed inso far that the total quantity is preserved, but they are also mutations.For more general morphological objects, such as the line R the notionof quantity is not defined and we will illustrate that, if it were definedit wouldn’t correspond to the cardinality of a set.Yet we calculate as if these objects would have a form of quantity andso, in particular, terms in an addition may be commuted, substitutedand brackets may be put or deleted.For the multiplication v · w , commutativity v · w = w · v is even less obvi-ous especially if one thinks of a fibre bundle E = M · F . But again thesegeometrical interpretations happen outside morphological calculus andthe total quantity of v · w is the same as that of w · v . Moreover, to beable to calculate one has to be able to commute factors in a product,xamples of Morphological Calculus 5even though this deforms the morphological structure. Also the law ofdistributivity( t + · · · + t k ) · w = t · w + · · · + t k · w is essential to give a meaning to the product while as the same time itis a deformation.So, to conclude, the morphological universe consists of the totality of allmeaningful algebraic expressions based on a set of names for morpholog-ical objects together with their definitions within calculus. The calculusrules, leading to the relations A = B are the same as for the naturalnumbers and the relations A = B are interpreted at the same time asmorphological deformation and as preservation of quantity, whatevermeaning this may have.
2. The Real Line
The real line R is in mathematics defined as the set of all real numbers,represented as points on that line. It is hence an infinite point set and itscardinality c is called the continuum; it is larger than the cardinality ℵ ofthe natural numbers.The real line decomposes as R = R − ∪ { } ∪ R + with R − = ] − ∞ ,
0[ : the halfline of negative numbers. R + = ]0 , + ∞ [ : the halfline of positive numbers.So R + and R − are open intervals that are closed off and glued together bythe point { } to form the real line.Morphologically we write this disjoint union as R = R − + 1 + R + whereby “1” represents the middle point { } .Both R + and R − are halflines having “the same shape”, so we identify R − = R + , leading to the first definition R = R + + 1 + R + , which, after commuting terms, leads to R = 2 R + + 1 . Next one may argue that all open intervals ] a, b [ have “the same shape”, sothey are all copies of R and, in particular, we may identify R + = R , leading to the relations R = R + 1 + R = 2 R + 1 . Frank SommenThis may be interpreted as the way to produce an open interval or curve ] a, c [by taking an open interval ] a, b [, glue to it a point { b } and then glue to thenext open interval or curve ] b, c [.The question now is: what is the quantity of R ?If it is the cardinality “ c ” then one should identify R + 1 with R but R + 1would be a semi-interval like ]0 , , R = R + 1 + R indicates the fact that R contains at leastone point and, by iteration R = R + 1 + R = ( R + 1 + R ) + 1 + ( R + 1 + R ) = · · · we obtain 3 points, 7 points, 15 points etc., any finite number of points. Sothe morphological version of R seems to house infinity many points.Now let us consider the relation R = R + 1 + R = 2 R + 1as an equation. Then by subtracting R from both sides we get R + 1 = 0and by subtracting 1 we get R = 0 − − , so that the total quantity of R should be − R being a set of points; the morpho-logical line is hence not merely a set of points but rather a brand new objectthat doesn’t quantify as a pointset. Of course one could argue that alsoinfinity = 2infinity + 1 , but infinity is a too trivial and vague number to work with for it absorbseverything.There is an interesting interpretation for R = − . Every manifold or surface of finite dimension may be represented by a cellcomplex, which we may represent by a polynomial a o R n + a R n − + · · · + a n , a ∈ N , a , . . . , a n ∈ N ∪ { } . By making the identification R = − e { M } = a ( − n + a ( − n − + · · · + a n , which is the Euler characteristic e { M } of manifold M .The Euler number e { M } is a topological invariant and for a given manifold M it is independent of the cell decomposition of that manifold. To see this,note that for any two cell decompositions of M there exists a kind cell de-composition that refines both of them and so it suffices to consider the case M = R n . Moreover, every cell decomposition of R n may be obtained fromsimple cell decompositions of the form R j = 2 R j + R j − , which proves theinvariance of e { M } morphologically.The fact that morphological calculus respects the Euler characteristicis like a corner stone (it is the final invariant that is preserved!). But as it isxamples of Morphological Calculus 7now, morphological calculus is reduced to the calculus of the integers Z anda point 1 is identified with a closed interval R + 2 a plane R is identifiedwith a point 1.Hence, the idea of a line as an infinite point set is completely lost andalso the dimension of an object is not preserved. As a result we have theidentification R + 1 = 0 between a semi-interval (or circle) R + 1 and thenumber zero and, in fact R = − R and the number −
1, whilenumbers 0 and − Axiom . Morphological calculations are only granted if all the algebraicexpressions and operations make sense in terms of geometrical objects.Hence, in particular, number zero 0 and negative numbers − , − , etc.are hereby excluded or at least pushed to the background. Moreover, a rela-tion like R = R + 1 + R = 2 R + 1does not automatically allow one to solve it like an equation; one could alsosimply interpret it by stating that one is allowed to replace R by 2 R +1 or vice-versa within calculations and nothing more. Hence, it does not automaticallyimply e.g. that R + 1 = 0 or even R − ≡ R , although this last relation R − R makes morphological sense. This now leads to the following result. Theorem 2.2 (Morphological Stability).
Under the assumption of the relation R = 2 R + 1 , every cell complex a R n + a R n − + · · · + a is equivalent toeither a R n , a ∈ N or R n + b R n − , b ∈ N , no further identifications being possible.Proof. The statement holds trivially for n = 0. For n = 1, a > b > a R + b = ( a − R + 2 R + ( b −
1) + 1 = ( a − R + b − b = 0 or a = 1 . Next, assuming the property for n − n >
1, we may reduce any cell complexto a R n + b R n − + c R n − . If c = 0 we may reason as in the case n = 1 to arrive at the final form a R n or R n + b R n − . If c > a R n + b R n − + c R n − = ( a − R n + (cid:0) R n + R n − (cid:1) + b R n − + c R n − = ( a + 1) R n + ( b + 1) R n − + c R n − and repeat this idea until b > . Then one may reduce using 2 R + 1 = R : a R n + b R n − + c R n = a R n + ( b − R n − + ( c − R n − and so on, until thefinal form is reached. (cid:3) Note that this theorem guarantees us that in the worst case, at leastthe dimension = n Frank Sommenas well as theEuler characteristic = ( − n a or ( − n + ( − n − b are being preserved during morphological calculations; it is a second approxi-mation for any possible notion of morphological quantity (the first one beingjust the Euler characteristic).But this calculus is still too poor and to be able to evaluate the quantity of R we have to ignore the distinction between a line R and a halfline R + . Thisis again a dilemma, similar the once mentioned in introduction concerningcommutativity and use of brackets. There are two options1. The “canonical” option.Hereby we assume as definition for R the relation R = R + + 1 + R + = 2 R + + 1and consider the identification R + = R as a form of decay. So the re-lation R = 2 R + 1 is suspended in what we regard as “the canonicalstyle”.This style of calculating is on the other hand flexible with respect tocommutativity and the use of brackets. Its main purpose is (not exclu-sively): Morphological analysis: to analyse geometrical objects (surfaces,manifolds) by decomposing them into parts (or other ways) and to ex-press this knowledge in calculus language in order to arrive at morpho-logical definitions.2. The “formal” option.Hereby we consider morphological calculus as a formed language inwhich the order of terms in an addition and the use of brackets is notignored. For the morphological line we have two definitions: R = R + 1 + R or R = R + ( R + 1) . Its main purpose is (not exclusively):Morphological synthesis: to construct a geometrical interpretation foran algebraic expression in morphological calculus.In this paper we mostly use the canonical style. Our main interest is to studymanifolds and try to understand their morphological quantity, whatever thatmay mean. The formal style will be discussed briefly in last section.
3. Carthesian Space, Spheres, Projective Spaces
The carthesian plane is defined as the product R = R · R , using the relation R = 2 R + + 1 we thus arrive at R = (2 R + + 1) = 4 R + 4 R + + 1 , xamples of Morphological Calculus 9decomposing the plane into 4 quadrants R , R + and one point1 (the origin).Similarly the cartesian n -space is defined as product R n = R · · · · · R = R · R n − and we have its decomposition into “octants”: R n = (2 R + + 1) n = n X j =0 (cid:18) nj (cid:19) j R j + . To define the sphere S n − we make the following analysis: for a vector x ∈ R n with x = 0 we have the polar decomposition x = rω , r = | x | ∈ R + , ω = x | x | ∈ S n − .In morphological calculus language we write R n − S n − R + , leading to the morphological definition of S n − : S n − = R n − R + . Now from R = 2 R + + 1 we obtain that R + = R − , a line without a point indeed gives 2 halflines.Hence we obtain a quantity formula for S n − : S n − = 2 R n − R − R n − + 2 R n − + · · · + 2 R + 2 . In particular, a circle is given by S = 2 R + 2 : two semi-circles and two points and a 2-sphere is given by S = 2 R + 2 R + 2 = 2 R + S , two hemi-spheres and a circle (equator). Also S = S R + 2 , a “cylinder S R ” and two poles “2”.Using here R = 2 R + + 1 we obtain S = 2 (2 R + + 1) + 2(2 R + + 1) + 2 = 8 R + 12 R + + 6 , which may be interpreted as an octahedron whereby R translates as a triangle, R + translates as a quarter circle or short interval.Other regular polyhedra are harder to obtain, yet they are obtainable bytransformations of the form R = 2 R + 1 , R = 2 R + R , which as weknow are questionable. In fact, every cell complex a R + b R + c that corre-sponds to an embedded connected 2-manifold in R has Euler characteristic a − b + c = 2(1 − g ), g being the genus or number of holes, a number whichcharacterises the manifold. Hence, for 2 − manifolds the relation R = 2 R + 10 Frank Sommenis not such a destructive deformation.However, this also means that e.g. a dodecahedron will be identified with12 R + 30 R + 20 and hence a solid pentagon is identified with a square R ,an identification which is only topologically true.For general spheres we have the recursion formula S n − = 2 R n − + S n − : two hemi-spheres and an equator,as well as the “polar coordinate” formula S n − = S n − R + 2: a cylinder and 2 poles.These formula are special cases of the following general method for introduc-ing polar coordinates on S n − .Let ω ∈ S n − and consider the decomposition R n = R p × R q , p + q = n . Thenwe may write ω = cos θω + sin θω , θ = h , π i , ω ∈ S p − , ω ∈ S q − . There are three cases: θ = 0 : ω = ω ∈ S p − , θ = π ω = ω ∈ S q − ,θ ∈ i , π h : ω = cos θω + sin θω ∼ ( ω , ω ) ∈ S p − × S q − . In morphological calculus this situation is expressed as follows: R n − R p −
1) ( R q −
1) + ( R p −
1) + ( R q −
1) or S n − R + = (cid:0) S p − R + (cid:1) (cid:0) S q − R + (cid:1) + (cid:0) S p − R + (cid:1) + (cid:0) S q − R + (cid:1) leading to the addition formula for spheres: S n − = S p − · S q − · R + + S p − + S q − . Notice that also here R + is interpreted as the quarter circle (small interval) θ ∈ (cid:3) , π (cid:2) , while the full line R would rather correspond to a semi-circle θ ∈ ]0 , π [.The addition formula also leads to: S n − = S p − (cid:0) S q − R + + 1 (cid:1) + S q − = S p − R q + S q − , which generalises the recursion formula and the “polar coordinate” formulamentioned earlier.Of particular interest is the odd-dimensional sphere S n − where we can take p = q = n .This leads to the “Hopf factorization formula” S n − = S n − R n + S n − or S n − = ( R n + 1) S n − . In particular we have the Hopf fibrations S = S S , S = S S that are well known and follow from complex resp. quaternionic projectivegeometry. They can be seen as interpretations of the Hopf factorization S = (cid:0) R + 1 (cid:1) S , S = (cid:0) R + 1 (cid:1) S , xamples of Morphological Calculus 11whereby the spheres S resp. S are identified with (cid:0) R + 1 (cid:1) resp. (cid:0) R + 1 (cid:1) .But of course the Hopf fibrations are by no means proved or even implied bymorphological calculus.In general, the sphere S n − can be mapped onto R n − by stereographic pro-jection. Hereby one takes line from the southpole w = (0 , . . . , , −
1) to generalpoint w , denoted by L ( w ) and the stereographic projection st( w ) is the in-tersection of L ( w ) with plane x n = 1 (the tangent plane to the north pole(0 , . . . , , +1)).This leads to the identification between S n − and R n − ∪ {∞} . In morpho-logical calculus one might hence think of identification S n − = R n − + 1. Butthat would lead to the unwanted identification2 R + 2 R + 2 = R + 1 , that would again correspond to R = 2 R + 1 via: R + 1 = (2 R + 1) R + 1 = 2 R + R + 1 = 2 R + (2 R + 1) + 1 = 2 R + 2 R + 2 . In morphological calculus we introduce a kind of stereographic sphere or“Poincar´e sphere” by S n = R n + 1 , leading to the recursion formula S n = (2 R + + 1) R n − + 1 = 2 R n − R + + S n − and leading to the total quantity (whatever that may mean) S n = 2 R n − R + + 2 R n − R + + · · · + 2 R + + 2 . Notice hence that the identification R + = R would lead to S n = S n or S = R + 1 (a point and a square is a sphere). It is true that the only2 − manifold interpretation for R + 1 is indeed a sphere. Also the Poincar´e-polynomial of the sphere S n is given by t n + 1, which corresponds to R n + 1.Recall that the Poincar´e-polynomial of a manifold M is defined as a n t n + · · · + a with a j = dim H j , H j leading the j − th homology space of M .It turns out that the Poincar´e-polynomial often appears as the morphologicalquantity of an object, in particular for R n itself and the sphere S n . But for thesphere S n we obtain a “higher” morphological quantity: 2 R n + · · · + 2 R + 2that does not correspond to the Poincar´e-polynomial. The real projectivespace RP n corresponds to the set of 1 D subspaces of R n +1 , also defined asthe set of vectors ( x , . . . , x n +1 ) ∼ ( λx , . . . , λx n +1 ) , λ = 0 . In mathematics we write it as the quotient structure RP n = R n +1 \{ } R \{ } . This leads to the morphological definition RP n = R n +1 − R − . and to the formula for the quantity of RP n : RP n = R n + R n +1 + · · · + R + 1 . RP n : t n + · · · + t + 1 . It also leads to the recurson formula RP n = R n + RP n − in which “ R n ” symbolizes the Affine subspace consisting of the points ( x , . . . , x n , RP n − ” stands for the plane at infinity: x n +1 = 0 . Of course we also have that RP n = S n S = S n S in the multiplicative group S = {− , } .In particular the projective line is given by RP = R + 1symbolizing R ∪ {∞} and it also represents the Poincar´e circle S = R + 1 = S / ,S = 2 R + 2 being the standard circle.The projective plane is given by RP n = R + R + 1 = R + RP = ( R + 1) R + 1 , whereby the object ( R + 1) R in this context corresponds to a Moebius band.Just seen by itself, ( R + 1) R could correspond to several things, including anyline bundle over the circle R + 1, i.e., either a cylinder or a Moebius band. InGeometry the Moebius band can be recognised by cutting it in half along thecenter circle; if it was a cylinder, then the cutted object would give 2 cylindersand if it was a Moebius band then the cutted object would be a single cylinder.Now, this cutting procedure can be translated into morphological calculus asthe subtraction( R + 1) R − ( R + 1) = ( R + 1)( R −
1) = R − R − R − S R + = (2 R + 2) R + , here represented as a product of a circle 2 R + 2 (which has two glueing pointsand twice the length of the original circle) with a halfline R + (stretching fromthe cutting point { } to the boundary {∞} ).So this simple calculation symbolizes quite well the whole cutting experimentand it illustrates us the object ( R + 1) R as being a Moebius band. In general,morphological objects are merely organised quantities that can have a numberof meanings called morphological synthesis. This synthesis takes place outsidethe calculus but it can be guided by calculations that give the object anintrinsic meaning. In the Moebius experiment we also see that the circle2 R + 2 and the Poincar´e circle R + 1 clearly play different roles like also theline R and the halfline R + .If we apply a similar experiment to the cylinders(2 R + 2) R − (2 R + 2) = (2 R + 2)( R −
1) = 2 S R + xamples of Morphological Calculus 13we obtain two cylinders. Of course one always calculates in a certain way andthat may force a certain interpretation; the language of calculus can be usedas an illustration but not as a real proof. In fact the language of calculus alsohas to remain flexible enough but this flexibility is at the cost of the stabilityof the morphological-synthesis. For example we have R − S R + = (2 R + 2) R + = 2( R + 1) R + = 2 S R + showing that distributivity results in the cutting and reglueing of one cylinder S R + into two cylinders S R + , half the size and with one single cutting edge R + . For the general projective space we have a kind of “Moebius factorization” RP n = RP n − · R + 1whereby the Moebius cutting experiment is represented as RP n − · R − RP n − = RP n − ( R −
1) = R n − S n − R + , also a kind of cylinder.We now turn to complex projective spaces.The complex numbers C are morphologically given by C = R and this is all. Anything concerning √− i exists outside the calculus. Wealso have that C n = (cid:0) R (cid:1) n = R n . Complex projective space CP n is defined as the set of equivalence classes ofrelation ( z , . . . , z n +1 ) ∈ C n +1 \ { } ∼ ( λz , . . . , λz n +1 ) , λ ∈ C \ { } i.e. the quotient structure CP n = C n +1 \ { } / C \ { } . Hence, in morphological calculus we have the definition CP n = C n +1 − C − CP n = C n + C n +1 + · · · + 1 = R n + R n − + · · · + 1that also corresponds to Poicar´e polynomial. The Euler number of CP n equals n .We also have that in real terms: CP n − = R n − R − S n − S , leading to the CP n − factorization of S n − S n − = CP n − · S , z , . . . , z n ) ∈ S n − → (cid:0) e iθ z , . . . , e iθ z n (cid:1) .In particular we have that CP = C + 1 = R + 1 = S and the above fibration leads to the first Hopf-fibration S = S · S . Like in the real case one has the recursion formula CP n = C n + CP n − and also the Moebius factorization CP n = CP n − · C + 1 . Hereby the complex line bundle CP n − · C reduces for n = 2 to CP = S · C = ( C + 1) C and it is a non-trivial plane bundle over the 2 − sphere.In fact also here we have “Moebius cutting experiment” CP · C − CP = ( C + 1)( C − C − R − S R + . showing that fibration S ( C −
1) is non trivial: S R + .This remains true in general: CP n − · C − CP n − = CP n − ( C −
1) = C n − R n − S n − · R + . The above may be repeated for the quaternions; we present the morphologicalheadlines:We have H = R , H n = ( R ) n = R n , HP n = H n +1 − H − H n + H n − + · · · + 1 = R n + R n − + · · · + 1= H n + HP n − . Also HP n − = R n − R − S n − S leading to the HP n -factorization (fibration) S n − = HP n − · S , which in particular for n = 2 leads to the second Hopf-fibration S = HP · S = ( H + 1) S = ( R + 1) S = S S . The Moebius factorization is given by HP n − HP n − · H while we also have the Moebius cutting experiment: HP n − · H − HP n − = HP n − · ( H −
1) = H n − R n − S n − · R + . xamples of Morphological Calculus 15But not every interesting quotient in calculus leads to a morphological syn-thesis that produces a nice manifold. Yet these quotients are also interestingbecause they say a lot about the meaning of morphological calculus and wecall them “phantom geometrical objects”. Example. RP nh RP nh = R n +1 + 1 R + 1 = R n − R n − + · · · + R + 1 , which we call the Phantom (real) projective space of dimension 2 n. The simplest case is RP h = R − R + 1 , with Euler characteristic 3. This would be one too high for a connected2 − manifold and R − R + 1 corresponds to: take plane R , delete line R and add point 1; it makes sense as a weird object but not as a 2-manifold.In fact one could say R − R + 1 = (2 R + + 1) R − R + 1 = 2 R + R + 1 , two halfplanes (or half-discs or triangles) glued together by a single point (abutterfly).Note that we also have that RP h = S S . If we would now use R = 2 R + 1 we could make the identification S = S , S = S and arrive at S S = S S = S = R + 1 ( Hopf f ibration )and therefore R − R + 1 = R + 1 . This is total nonsense because this identification is even wrong on the levelof Euler numbers : 3 = 2 . The reason why such bad identification happens is because the Eulernumbers of S , S , S , S are all equal to zero, so, on the level of Euler num-bers: S S = 00 & S S = 00 , so one would not even be allowed to consider the quotients S /S , S / S . Butthat would also exclude CP n from the picture as well as the Hopf fibration, anunpermitable exclusion. This is a sound reason why the relations R = 2 R + 1or R + = R or S n = S n must be forbidden: they simply spoil the calculus.The general Phantom projective space RP nh = R n − R n − + · · · − R + 16 Frank Sommensurely makes sense as a geometrical object, but the corresponding quantity R n − R n − + · · ·− R + 1 still has negative numbers as coefficients, so it is notyet fully evaluated. This can be done by replacing R = 2 R + + 1 at suitableplaces, giving rise to R n − R n − + · · · − R + 1 = 2 R + R n − + 2 R + R n − + · · · + 2 R + R + 1 , which also provides a synthesis for RP nh . Comparing R n − R n − + · · ·− R +1with Poincar´e polynomial also suggests that some of the homology spaces of RP nh would have negative dimension. But we also have that phenomena withthe object R − R − R + 1) = 2 R + R + 2 R + . Quantity simply doesn’t always have a positive evaluation as an addition ofpowers R s . This leads to Definition 3.1.
A morphological object is called integrable if it has an evalu-ation of the form F = a R n + a R n − + · · · + a n , a ∈ N , a , . . . , a n ∈ N ∪ { } ;this polynomial is then called the “total quantity” or “integral”. The object F is called semi-integrable if it has an evaluation as an addition of terms ofthe form R j + R k . Such expression is not unique unless we require the power“ j ” of R + to be minimal, in which case the obtained expression is also called“total quantity” or “integral”.Note that not every object is semi-integrable; for example F = R − R and R + . One option would be to introduce newtype of line, e.g. R + = 2 R ++ + 1 , but that would not lead to be more interesting calculus.Notice that the Phantom projective plane can also be interpreted as theresult of the cutting experiment RP nh = R n − R n − + · · · − R + 1= (cid:0) R n + R n − + · · · + 1 (cid:1) − (cid:0) R n − + · · · + 1 (cid:1) R = CP n − CP n − · R . We also have the Phantom Moebius strip RP nh − R − CP n − · R and this time we have a Moebius “glueing-experiment”( R − CP n − · R + ( R − CP n − = CP n − ( R − R n − S n − · R + , xamples of Morphological Calculus 17the same cylinder as we had earlier on.Of course one may also consider complex and quaternionic phantom projec-tive spaces: CP nh = C n +1 + 1 C + 1 = C n − C n − + · · · − C + 1 = · · · , HP nh = H n +1 + 1 H + 1 = H n − H n − + · · · − H + 1 = · · · The fact that the corresponding synthesis for phantom projective spaces doesnot add up to a manifold implies that these quotients do not correspond toa group action (or else the quotients would be homogeneous spaces). Indeed,the denominators in the definition of the projective spaces are the multi-plicative groups R − , C − , H − S = R + 1 , S = C + 1 , S = H + 1 which are non-groupsleading to non-group actions. In fact group actions can not be recognisedwithin morphological calculus itself, only by the outside interpretations. Theconsideration of phantom geometry also leads to the next definition. Definition 3.2.
A morphological object is said to be of “integer type” if it hasan evaluation of the form F = a R n + a R n − + · · · + a n , a ∈ N , a , . . . , a n ∈ Z . A semi-integrable object that is not of integer type is to said to be of “halfinteger type”. Other objects are “just another type”.Notice that R − R j + R k , j > R + 2) R + = R − R + 1) R + is only of half-integer type.This example confirms that it is a good idea to keep two circles S = 2 R + 2 , S = R + 1 in use rather than deciding that 2 R + 2 = 2( R + 1) is always apair of circles. Note that the object R + − − R isn’t even an object. So we have a kind of hierarchy that is quantity based. Example.
Phantom fibrationsWe already discussed Hopf factorization S n − = ( R n + 1) S n − which only for n = 2 and n = 4 leads to a true fibration: the Hopf-fibrations S = S S , S = S S . These fibrations in fact correspond to projective geometry and the fac-tors S and S are group actions.In the other cases like e.g. S = S S S S = ( R + 1) S = R S + S does lead to a synthesis of S and it is like a fibration still, but an irregularfibration that would not locally correspond to a Cartesian product, whencethe name “phantom fibration”.For the spheres S n − we also have repeated factorizations S = (cid:0) R + 1 (cid:1) S = (cid:0) R + 1 (cid:1) (cid:0) R + 1 (cid:1) ( R + 1) 2 ,S = (cid:0) R + 1 (cid:1) (cid:0) R + 1 (cid:1) (cid:0) R + 1 (cid:1) ( R + 1) 2 , and so on. If we apply non-associativity we get S = (cid:0)(cid:0) R + 1 (cid:1) (cid:0) R + 1 (cid:1)(cid:1) S = CP S ,S = (cid:0) R + 1 (cid:1) (cid:0)(cid:0) R + 1 (cid:1) ( R + 1) 2 (cid:1) = (cid:0) R + 1 (cid:1) S = S S , two fibrations of S that follow from complex and quaternion geometry andthat are unrelated.There are also more Hopf factorizations, the simplest one being S = (cid:0) R + R + 1 (cid:1) S . In general they follow from products of the form (cid:16) R s · k + R ( s − · k + · · · + R k + 1 (cid:17) (cid:0) R k − + 2 R k − + · · · + 2 (cid:1) leading to S ( s +1) k − = (cid:0) R s · k + · · · + R k + 1 (cid:1) S k − and they play a crucial role in the “Graßmann division problem”.Needless to say that there are repeated factorizations of this type.Also the addition formula for spheres may be generalised.For p + q + r = m we have R m − R p −
1) ( R q −
1) ( R r −
1) + ( R p −
1) ( R q − R p −
1) ( R r −
1) + ( R q −
1) ( R r −
1) + ( R p −
1) + ( R q −
1) + ( R r − , from which we obtain: S m − = S p − S q − S r − R + S p − S q − R + + S p − S r − R + + S q − S r − R + + S p − + S q − + S r − . Needless to say also that our list of interesting manifolds and geometries isfar from complete.Let’s take the Klein bottle as an example, we have the following mor-phological analysis. A Klein bottle can be obtained from a Moebius bandby properly glueing a circle to the edge, thus closing it up into a compact2-manifold. As we know, a Moebius band may be obtained by removing apoint from RP : RP − . Then one blow up the hole to a small disc andone glues a circle S = 2 R + 2 to that, giving 2-manifold with boundary.Finally one identifies every point on this S with its anti-podal point: S / Z xamples of Morphological Calculus 19which leads to a continuation across the boundary and to the Klein bottle.In morphological language we have: (cid:0) RP − (cid:1) + S / Z = (cid:0) RP − (cid:1) + ( R + 1)= (cid:0)(cid:0) R + R + 1 (cid:1) − (cid:1) + ( R + 1)= (cid:0) R + R (cid:1) + ( R + 1)= ( R + 1) R + ( R + 1) = ( R + 1) ( R + 1) , so we end up with a circle S -bundle over S . But S · S may also simplyrepresent a torus: there is no way one can tell from the quantity ( R + 1) alone whether this represents a torus or a Klein-bottle. Only in the initialformula (cid:0) RP − (cid:1) + ( R + 1) one can specify a Klein bottle but as one startscalculating, this specification is lost.Higher dimensional Klein bottles may be introduced as the “blow up” exper-iment: ( RP n −
1) + S n − / Z = ( RP n −
1) + RP n − = RP n − · R + RP n − = RP n − · S , an S -bundle over RP n − .Similarly, complex and quaternionic Klein-bottles may be introduced as (ex-ercise) the “blow up experiment”:( CP n −
1) + S n − /S = CP n − · ( C + 1) , ( HP n −
1) + S n − /S = HP n − · ( H + 1) . To summarise this section, we notice that there is no one to one correspon-dence between morphological calculus and geometry. This may be seen asa drawback but it is also a stronghold because it means that there existsanother perspective that reveals a hidden aspect of geometry: the quantityof an object.
4. Groups and Homogeneous Spaces
Groups enter morphological calculus via a proper morphological analysis; thegroup structure will be lost and the organised quantity remains.We begin with the groups O ( n ), SO ( n ), GL ( n, R ), GL ( n, R ), SL ( n, R ).The orthogonal group O ( n ) is the group of all orthogonal matrices ( a ij ). Ifwe represent such a matrix as a row (a¯ , . . . , a¯ n ) of column vectors it simplymeans that a¯ , . . . , a¯ n are orthogonal unit vectors. This means that one canstart off by choosing a¯ ∈ S n − followed by choosinga¯ ∈ S n − ∩ { λ a¯ , λ ∈ R } ⊥ = S n − and then a¯ ∈ S n − ∩ { λ a¯ + λ a¯ , λ j ∈ R } ⊥ = S n − n there are just 2 choicesa¯ n ∈ S n − ∩ span { a¯ , . . . , a¯ n − } ⊥ = S . This immediately leads to the morphological definition O ( n ) = S n − · S n − · · · S , as well as to the recursion formula O ( n ) = S n − · O ( n − , O (0) = 1 . For the group SO ( n ) everything remains the same except that for the lastvector a¯ n there is just one choice, determined by det( a ij ) = 1 condition.We thus have the definition SO ( n ) = S n − S n − · · · S = O ( n ) / Z . Clearly O ( n ) , SO ( n ) are integrable and the integral is obtained by substitut-ing S j − = 2 R j − + · · · + 2 and working out the product.The general linear group GL ( n, R ) is obtained similarly by writing matrix( a ij ) as (a¯ , . . . , a¯ n ) wherebya¯ ∈ R n \ { } , a¯ ∈ R n \ span { a¯ } , ...a¯ n ∈ R n \ span { a¯ , . . . , a¯ n − } which leads to the morphological definition GL ( n, R ) = ( R n −
1) ( R n − R ) · · · (cid:0) R n − R n − (cid:1) . We readily obtain the quotient formula GL ( n, R ) O ( n ) = ( R n −
1) ( R n − R ) · · · (cid:0) R n − R n − (cid:1) S n − · S n − · · · S = (cid:18) R n − S n − (cid:19) (cid:18) R n − R S n − (cid:19) · · · (cid:0) R n − R n − (cid:1) S = R + · ( R · R + ) · · · (cid:0) R n − · R + (cid:1) , which symbolizes the GRAMM-SCHMIDT orthogonalization procedure. Herewe applied commutativity of the product but that doesn’t matter too much;in fact one can also write GL ( n, R ) = (cid:0) S n − R + (cid:1) (cid:0) R S n − R + (cid:1) · · · (cid:0) R n − S R + (cid:1) . For the group SL ( n, R ) we have the extra condition det ( a ij ) = 1 , whichreadily leads to SL ( n, R ) = GL ( n, R ) R − , and so also SL ( n, R ) SO ( n ) = R + · ( R · R + ) · · · (cid:0) R n − · R + (cid:1) R n − . xamples of Morphological Calculus 21Now let us look some homogeneous spaces.The Stiefel manifold V n,k ( R ) is by definition the manifold of orthonormal k -frames (v¯ , . . . , v¯ k ) in R n . We hence have that for k < n : V n,k ( R ) = SO ( n ) SO ( n − k ) = S n − · · · S n − k = O ( n ) O ( n − k ) . The Stiefel manifold V n,k ( R ) is the manifold of k -frames (v¯ , . . . , v¯ k )that are linearly independent and hence span a k -plane. We have for k < n : V L n,k ( R ) = GL ( n, R ) R n − k · GL ( n − k, R ) = ( R n −
1) ( R n − R ) · · · (cid:0) R n − R k − (cid:1) . The Graßmann manifold G n,k ( R ) is the manifold of k -dimensional subspacesof R n . Now, each k -dimensional subspace has an orthogonal frame and thatcan be chosen in O ( k ) in different ways. This leads to the combinatorialformula: G n,k ( R ) = V L n,k ( R ) O ( k ) = O ( n ) O ( k ) · O ( n − k ) = S n − · S n − · · · S n − k S k − · · · S . The Graßmann manifold may also be constructed starting from the generallinear group: G n,k ( R ) = V n,k ( R ) GL ( k, R ) = ( R n −
1) ( R n − R ) · · · (cid:0) R n − R k − (cid:1) ( R k − · · · ( R k − R k − )and the equivalence of both definitions readily follows from the Gramm-Schmidt factorization.By ] G n,k ( R ) we denote the manifold of all ORIENTED k -dimensionalsubspaces of R n , i.e., ] G n,k ( R ) = V n,k ( R ) SO ( k ) = SO ( n ) SO ( k ) · SO ( n − k ) = S n − · S n − · · · S n − k S k − · · · S . Now, for the Stiefel manifolds everything is clear, but for the Graßmannmanifolds we have one major problem.
Problem . Graßmann division problemCan one work out the polynomial division S n − · S n − ··· S n − k S k − ··· S , and does it resultin an integral (polynomial in “ R ” with natural number coefficients).To solve the problem we will work with the quotient V L n,k ( R ) /GL ( k, R )that is equivalent and easier to work with. For the case of simplicity take k = 3. Every 3 D -subspace V of R n is spanned by 3 linearly independentvectors v¯ , v¯ , v¯ that may be chosen in GL (3 , R ) different ways. For each V there is a unique triple (v¯ , v¯ , v¯ ) that may be written as a matrix of theform v¯ v¯ v¯ = c · · · c j c j +2 · · · c j c j +2 · · · c j · · · c · · · c j c j +2 · · · c j · · · · · · c · · · c j · · · · · · · · · V may be obtained by a unique GL (3 , R )-actionfrom this, so in fact the division is carried out by looking to matrices of theabove special form. As the coefficients c ij vary the matrices of the above formconstitute a cell of G n, ( R ) that is a copy of a certain R j and it is called aSchubert cell. We thus have proved the following results. Theorem 4.2.
Schubert cellsThe object G n,k ( R ) = R d + c R d − + · · · + c d whereby c : j ∈ N is the numberof Schubert cells of dimension d − j. Apart from this there are typical morphological questions such as: . Q1: To decompose G n,k ( R ) = O · · · O s as a (e.g. maximal) product ofmorphological objects O j of integer type (that are irreducible e.g.). . Q2: To look for G n,k ( R )-factorization O · · · O t in terms of objects O j that are integrable.Let us consider a few examples of such Graßmann factorizations.Of course we readily have G n,k ( R ) = RP n − and the Hopf factorizationsprovide further ways of factorizing this.Next for k = 2 we have: G n, ( R ) = S n − · S n − S · S = CP n − · RP n − G n +1 , ( R ) = S n · S n − S · S = RP n · CP n − , showing a clear 2 − periodicity.For k = 3 the first interesting case is G , ( R ) = S · S · S S · S · S , which, using the Hoft factorizations S = (cid:0) R + 1 (cid:1) S = S S , S = (cid:0) R + 1 (cid:1) S = S S may be evaluated as G , ( R ) = (cid:0) R + 1 (cid:1) · RP · (cid:0) R + 1 (cid:1) = RP · S · S . Note here that it is forbidden to divide S /S = 1 . More interesting still is the next case G , ( R ) = S · S · S S · S · S , which, using the Hopf-factorization S = ( R + 1) S yields. G , ( R ) = RP · (cid:0) R + 1 (cid:1) RP ( R + 1) . Now RP and RP cannot be divided by ( R + 1); in fact these objects areirreducible in morphological sense. So, the division that works here is: RP h = R + 1 R + 1 = R − R + 1 , xamples of Morphological Calculus 23the phantom projective plane, leading to the following maximal factorization G , ( R ) = RP · RP · RP h in terms of irreducible objects of integer type.But now the factors are no longer integrable, which also shows that theintegrability of Graßmann manifolds is in fact not so trivial. But we have: (cid:0) R + 1 (cid:1) R + 1 RP = (cid:0) R + 1 (cid:1) R + 1 R (cid:0) R − (cid:1) R − R + 1) ! = R (cid:0) R − (cid:1) R − (cid:0) R + 1 (cid:1) = CP · R + S , so that in fact we have integrable factorization G , = RP · (cid:0) CP · R + S (cid:1) . The next case is again simpler: G , ( R ) = S · S · S S · S · S = (cid:0) CP · RP (cid:1) (cid:0) R + 1 (cid:1) . For the next case G , ( R ) = S · S · S S · S · S , we have to use the next Hopf factorization S = (cid:0) R + R + 1 (cid:1) S , which gives us: G , ( R ) = (cid:0) R + R + 1 (cid:1) CP · RP . The next cases are: G , ( R ) = S · S · S S · S · S = CP · (cid:0) R + R + 1 (cid:1) · RP , the first appearance of an odd dimensional RP n , and G , ( R ) = S · S · S S · S · S = RP · CP · (cid:0) R + R + 1 (cid:1) . In the next case we again have 2 odd spheres and the Hopf factorization S = (cid:0) R + 1 (cid:1) (cid:0) R + 1 (cid:1) S = S · S · S , giving rise to G , ( R ) = S · S · S S · S · S = RP · CP · S · S . and finally in the next case we again have two irreducible spheres S , S ,leading to G , ( R ) = S · S · S S · S · S = RP · RP · S (cid:0) R + 1 (cid:1) R + 1 . where once again, the phantom projective plane appears R + 1 R + 1 = R − R + 1 = RP h . k = 3 . The formulas obtained here lead to a classification but they do notcorrespond to the fibre bundles of any kind. Besides, we used repeatedly thefact that quantity is commutative. Another interesting homogeneous spaceis Flag Manifold F n ; k,ℓ ( R ), k < ℓ < n whereby W is subspace of R n ofdimension 1 and V ⊂ W is a subspace of dimension k . This clearly leads tothe fibration F n ; k,ℓ ( R ) = G n,ℓ ( R ) · G ℓ,k ( R )= O ( n ) O ( n − ℓ ) O ( ℓ ) · O ( ℓ ) O ( ℓ − k ) · O ( k ) = O ( n ) O ( k ) O ( ℓ − k ) · O ( n − ℓ ) . The flag manifold F n ; k,ℓ ( R ) may also be seen as manifold ( V, V ′ ) with V ⊂ R n a subspace of dimension k and V ⊥ V ′ of dimension ℓ − k. The link withthe previous definition simply follows from W = V ⊕ V ′ and we have thefibration F n ; k,ℓ ( R ) = G n,k ( R ) · G n − k,ℓ − k ( R )= O ( n ) O ( k ) O ( n − k ) · O ( n − k ) O ( ℓ − k ) · O ( n − ℓ ) = O ( n ) O ( k ) O ( ℓ − k ) · O ( n − ℓ ) . More in general for 0 < k < . . . k s < n we may define the flag manifold F n ; k ,...,k s ( R ) as the manifold of flags ( V . . . , V s ) with V ⊂ V · · · ⊂ V s ⊂ R n subspaces of dimension dim V j = k j . We clearly have the iterated fibration F n ; k ,...,k s ( R ) = G n,k s ( R ) · G k s ,k s − ( R ) · · · G k ,k ( R )= O ( n ) O ( n − k s ) O ( k s ) · O ( k s ) O ( k s − k s − ) · O ( k s − ) · · · O ( k ) O ( k − k ) · O ( k )= O ( n ) O ( n − k s ) O ( k s − k s − ) · · · O ( k − k ) O ( k ) . Using orthogonal subspaces, we have: F n ; k ,...,k s ( R ) = G n,k ( R ) · G n − k ,k − k ( R ) · · · G n − k s − ,k s − k s − ( R ) . Orthogonal groups may also be defined for the spaces R p,q with pseudo-Euclidean inner product h x, y i = x y + · · · + x p y q − x p +1 y p +1 − · · · − x p + q y p + q . The corresponding groups are O ( p, q ) and SO ( p, q ). The group SO ( p, q ) e.g.is determined as the manifold of frames of signature ( p, q ) : (cid:0) v¯ , . . . , v¯ p ; v¯ p +1 , . . . , v¯ p + q (cid:1) whereby v¯ ∈ S p − · R q is the first spacelike vector v¯ ⊥ v¯ ∈ S p − · R q up tov¯ p ⊥ span (cid:0) v¯ , . . . , v¯ p − (cid:1) ∈ S · R q and the remaining vectors (cid:0) v¯ p +1 , . . . , v¯ p + q (cid:1) form a right oriented time-like q -frame, i.e. v¯ p +1 ∈ S q − , v¯ p +2 ∈ S q − andv¯ p + q is fixed by the fact that the determinant of the whole frame equals +1 . xamples of Morphological Calculus 25In total, the morphological bill adds up to: SO ( p, q ) = (cid:0) S p − · R q (cid:1) · · · (cid:0) S · R q (cid:1) S q − · · · S = O ( p ) · SO ( q ) · R p · q . and it is a two component group still.All of the above may be generalized to the complex Hermitian case. Letus start with C n provided with the Hermitian inner product:(z¯ , w¯ ) = z w + · · · + z n w n . Then by U ( n ) we denote the unitary group of matrices learning the Hermit-ian form invariant; its matrices may be written as Hermitian orthonormalframes v¯ , . . . , v¯ n whereby | v¯ j | = 1 , (v¯ j , v¯ k ) = 0 for j = k .This leads to the following morphological analysis:v¯ ∈ S n − is the unit vector in C n = R n ,v¯ ⊥ v¯ in hermitian sense, i.e. v¯ ∈ v¯ ⊥ ∩ S n − = S n − up tov¯ n ⊥ v¯ , . . . , v¯ n − , i.e. v¯ n ∈ S and, therefore, U ( n ) = S n − · S n − · · · S . In the above, please note that h v¯ , w¯ i = Re(v¯ , w¯ ) is the orthogonal innerproduct in R n and so(v¯ , w¯ ) = 0 iff h v¯ , w¯ i = 0 and h i v¯ , w¯ i = 0 . Clearly U ( n )is a subgroup of SO (2 n ) and for the quotient we have: SO (2 n ) U ( n ) = S n − ˙ S n − · · · S , which actually is a manifold, namely the manifold of all complex structureson R n (Exercise).The special unitary group SU ( n ) is the subgroup of matrices in U ( n )with determinant = 1, i.e., SU ( n ) = S n − · · · S , and in particular SU (2) = S . The definition of the complex general and special linear groups is obvi-ous; they are denoted by GL ( n, C ), SL ( n, C ). Like for the orthogonal groupsalso for the complex group U ( n ) we have the associated homogeneous spaces,in particular Graßmann manifolds G n,k ( C ) = U ( n ) U ( k ) · U ( n − k ) = S n − · · · S n − k +1 S k − · S k − · · · S , so for example G , ( C ) = S · S S · S = ( R + 1) CP = S CP = HP · CP .G , ( C ) = S · S S · S = CP · ( R + 1) = CP · HP . G , ( C ) = S · S S · S = HP · CP . and so we have again a clear 2-periodicity.We leave the discussion of G n, ( C ) as an exercise.Unitary groups may also be constructed over spaces C p,q with pseudo-Hermitian form( z, w ) = z w + · · · + z p w p − z p +1 w p +1 − · · · − z p + q w p + q and the corresponding invariance groups are denoted by U ( p, q ) and SU ( p, q )in case det = 1.The corresponding frames now have to be chosen on the pseudo-Hermitianunit sphere: | z | + · · · + | z p | − | z p +1 | − · · · − | z p + q | = 1which leads to the morphological formula U ( p, q ) = ( S p − · C q ) · ( S p − · C q ) · · · ( S · C q ) · S q − · S q − · · · S . Of course we also have the complexified versions O ( n, C ) and SO ( n, C ) of O ( n ) and SO ( n ); it is another story which we’ll leave out for the moment.To finish the list of matrix groups leading to morphological analysis, wemention the compact symplectic groups Sp ( n ); they follow from the quater-nionic Hermitian form ( q, r ) = q r + · · · + q n r n , whereby q j = q j + iq j + jq j + kq j is a quaternion and q j = q j − iq j − jq j − kq j its quaternion conjugate. Sp ( n ) is by definition the goup of quaternion n × n matrices leavingthis form invariant and its matrix elements may be regarded as quaternionicframes q , . . . , q n whereby q r ∈ H n with | q | = 1, i.e., q ∈ S n − , q ∈ H n with | q | = 1 and ( q , q ) = 0, i.e., q ∈ S n − , and so on. This leads to themorphological bill: Sp ( n ) = S n − · S n − · S , in particular Sp (1) = S and Sp (2) = S · S . Also here may be investigated quaternionic Graßmannians.The groups Sp ( n ) should not be confused with the non-compact groups Sp (2 n, R ) of matrices A ∈ GL (2 n, R ) leaving the maximal 2 − form invariant.For Sp (2 n, R ) we did not find a morphological evaluation yet.To finish this section we discuss the Spin groups Spin ( m ).We start by considering the real 2 m − dimensional Clifford algebra R m with generators e , . . . , e m and relations e j e k + e k e j = − δ jk . The space of bivectors R m, = X i,j b ij e i e j : b ij ∈ R xamples of Morphological Calculus 27forms a Lie algebra for the commutation product and the corresponding groupis the Spin group: Spin ( m ) = exp ( R m, ) . Its elements may be written into the form s = w · · · w s , w j = P w jk e k ∈ R m with w j = −
1, i.e., w j ∈ S m − .We have the following Spin ( m ) representation h : Spin ( m ) → SO ( m )whereby h : s → h ( s ) : x → sxs whereby for a ∈ R m , a is the conjugation with properties ab = b a & e j = − e j . In this way
Spin ( m ) is a 2 − fold covering group of SO ( m ), i.e., SO ( m ) = Spin ( m ) / Z and also Spin ( m ) is simply connected.This might suggest the morphological evaluation Spin ( m ) = SO ( m ) · Z = S m − · · · S · S = O ( m ) , which, through not entirely wrong in the sense of quantity, is somewhat un-interesting.But there is a more interesting evaluation of Spin ( m ).Let us start withSpin(3) = { q + q e + q e + q e : qq = 1 } = S = S · S = SU (2) = Sp (1)with differs rather substantially from O (3) = S · S · S . So in fact, the rotation group SO (3) has two different representationsin morphological calculus:one as the matrix group S O (3) = S · S and one in terms of the Spin group (quaternion S ): S O (3) =Spin(3) / Z = S / = S S / = S S = ( R + 1)( R + 1) = RP . In general we got S O ( m ) = Spin( m ) / , which is another morphological version of the rotation group.For m = 4 we consider the pseudoscalar e with e = +1 and e is central in the even subalgebra R +4 = Alg { e jk : j < k } ∼ = R ∼ = H ⊕ H . Putting E ± = 12 (1 ± e )8 Frank Sommenwe have E + + E − = 1 , E ± = E ± , E + E − = 0 , so every a ∈ R +4 may be written uniquely as: a = a + E + + a − E − , a ± ∈ H = span { , e , e , e } and in particular s ∈ Spin(4) : s + E + + s − E − , s ± ∈ S . So we have the morphological analysisSpin(4) =Spin(3) × Spin(3)= S · S = S · S · S . For m = 5, we use the fact thatSpin(5) = { s ∈ R +5 : ss = 1 } , together with the isomorphisms R +5 = R ∼ = H (2)i.e., the set of 2 × a = (cid:18) a a a a (cid:19) , a ij ∈ H and under this isormorphism we also have a = (cid:18) a a a a (cid:19) . This shows that in factSpin(5) = Sp(2) = S · S = S · S · S = S · S · S · S . For m = 6, the pseudoscalar e satisfies e = − R +6 = R , so it may identified with complex number i , leading to R +6 ∼ = C ⊗ R +5 ∼ = C ⊗ H (2) ∼ = C (4) , and under this map R +6 → C (4), the conjugate a of a ∈ R +6 corresponds tothe Hermitian conjugate ( a ) + of matrix ( a ) ∈ C (4).Hence the group G = { a : aa = 1 , a ∈ R +6 } corresponds to U (4) . Butfor m >
5, the group G no longer corresponds to Spin( m ) and for m = 6 G = exp nX b ij e ij + e o = exp nX b ij e ij o × exp { e } = Spin(6) × U (1)which shows that reallySpin(6) = SU (4) = S · S · S = S · S · S · S · S . For m = 7 on the situation is much more complicated. Could it be thatSpin(7) = S · S · S · S · S · S ?xamples of Morphological Calculus 29
5. Nullcones and Things
The nullcone
N C n − of complex dimension n − z , . . . , z n ) ∈ C n that satisfy z + . . . + z n = 0 . The complex ( n − C S n − consists of the solutions ( z , . . . , z n ) ofthe equation z + . . . + z n = 1.It is non-compact manifold that admits a canonical compactification C S n − ⊂ CP n given by the equation in homogeneous coordinates z , . . . , z n +1 : z + · · · + z n = z n +1 that is equivalent to z + · · · + z n + z n +1 = 0 if we replace z n +1 → i z n +1 . Thesubmanifold C S n − corresponds to the intersection with the region z n +1 =0 while the “points at infinity” corresponds to the intersection with plane z n +1 = 0 , leading to: C S n − : z + · · · + z n = 0 . Hence we have the disjoint union C S n − = C S n − ∪ C S n − . We are going to perform the morphological calculus of those objects in twodifferent ways, leading to two different formulas for the quantity (once again).The first method could be called the real geometry approach.Let us write z = x + i y , x = ( x , . . . , x n ), y = ( y , . . . , y n ) ∈ R n ;then the equation for N C n − may be rewritten as | x | = | y | & h x, y i = 0with | x | = x + · · · + x n , h x, y i = x y + · · · + x n y n . First solution is the point z = 0 with quantity 1.For z = 0 we may write x = ρω , y = ρν , ρ ∈ R + and ω, ν ∈ S n − such that ω ⊥ ν , i.e., ( ω, ν ) ∈ V n, ( R ). Hence we have N C n − = 1 + V n, ( R ) · R + = 1 + S n − · S n − · R + . The complex sphere C S n − written in real coordinates would lead to: | x | = 1 + | y | , h x, y i = 0 . First we have the case | y | = 0, | x | = 1 leading to the quantity S n − . Next for | y | ∈ R + we again may put x = rω , y = ρν whereby r = 1 + ρ , ρ ∈ R + and ω, ν ∈ S n − with ω ⊥ ν . This leads to the morphological bill C S n − = S n − + V n, ( R ) · R + = S n − + S n − · S n − · R + = S n − · (cid:0) S n − · R + (cid:1) = S n − · (cid:0) (cid:0) R n − − (cid:1)(cid:1) = S n − · R n − , which represents the tangent bundle to S n − . Once again remark that S n − · R n − might represent any ( n − S n − ormore general stuff, so it only represents the quantity of the tangent bundle.For C S n − we have two approaches. First it is the set of points ( z , . . . , z n +1 ) ∈ CP n solving the equation z + · · · + z n +1 = 0, which means that the homo-geneous coordinates ( z , . . . , z n +1 ) = 0 belong to N C n \ { } and they aredetermined up to a homogeneity factor λ ∈ C \ { } . This leads to C S n − = N C n − C − V n +1 , ( R ) · R + S · R + = V n +1 , ( R ) S = e G n +1 , ( R ) = S n · S n − S . Secondly we also have that C S n − = C S n − + C S n − S n − · R n − + S n − · R n − + · · · + S · R + 2giving the total quantity, while also C S n − + C S n − = S n − · (cid:18) R n − + S n − S (cid:19) S n − · (cid:0) (2 R + 2) R n − + S n − (cid:1) S = S n − · S n S , as expected.Note also that there is a 2-periodicity expressed by C S n − = S n · S n − S = S n · CP n − , C S n = S n +1 · S n S = CP n · S n . Note that we also have the identity
N C n − = 1 + C S n − · ( C − CS n .For CS we have the equation z + z = 0 ⇔ uv = 0 , u = z + i z , v = z − i z . Up to a factor λ = 0 there are solutions ( u, v ) namely (1 ,
0) and (0 , CS = 2 . For CS we have the equation uv = z including for z = 0 , uv = 0, i.e., CS and for z = 0 we normalise z = 1 , so we have the equation for CS : uv = 1 , i.e., u ∈ C \ { } , v = 1 /u . Thisleads to CS = C − CS = CS + CS = ( C −
1) + 2 = C + 1xamples of Morphological Calculus 31so in fact CS = CP = S . For CS we again have the splitting CS = CS + CS whereby CS is given by the equation uv = z − z = 1 − z , with normalization z = 1 . There are two cases of this: z = 1 , giving z ∈ C \ { , − } and z ∈ { +1 , − } . So, morphologically, we have a factor z ∈ C − z ∈ . In the case z ∈ C − uv = cte = 0to solve, which gives us u ∈ C − v = cte /u , leading to the quantity:( C − C − . For z ∈ uv = 0 to be solved, which gives us ( u, v ) = (0 , v = 0 and u ∈ C − u = 0 and u ∈ C − . So the total quantity is1 + 2( C − , with an extra factor 2, which gives the total CS = ( C − C −
2) + 4( C −
1) + 2= ( C − C + 2) + 2 = C + C = ( C + 1) C . Hence, we arrive at C S = ( C + 1) C + ( C + 1) = ( C + 1) = S CP = S S S . For CS we have the equation u v = 1 − u v leading to the cases u v = 1 and u v = 1 for which we have the morpho-logical factors C − C − C + 1 (the phantom complex projective plane).In case 1 − u v = c = 0 the remaining equation u v = c yields the factor C − u v we have u v = 0, i.e., 1 + 2( C − . In total thisgives CS = ( C − C − C + 1) + (1 + 2( C − C − C − C + C ) = ( C − C , which is also clear from the fact that u v + u v = 1 is basically the equation ad − bc = 1 for SL (2 , C ) = (cid:0) C − (cid:1) C . CS = CS + CS = (cid:0) C − (cid:1) C + ( C + 1) = ( C + 1) (( C − C + C + 1) = ( C + 1)( C + 1)= S S S = S · CP = CP . For CS we have the equation u v + u v = 1 − z leading to the factors C − z = 1 and 2 for z = 1.For 1 − z = c = 0 the remaining equation gives the factor SL (2 , C ) =( C − C while for c = 0 we have the equation u v + u v = 0, which is thenullcone N C = 1 + CS · ( C −
1) = 1 + ( C − C + 1) . In total we get CS = ( C − C ( C −
2) + 2( C − C + 1) + 2= ( C − C + 2) + 2 = C + C = C ( C + 1)so that CS = CS + CS = (cid:0) C + 1 (cid:1) (cid:0) C + C + 1 (cid:1) = CP · S = S S S . It seems that in general we will have CS n = S n +1 S S n = CP n · S n , CS n − = S n S n − S = S n · CP n − = CP n − . To prove this recursively we begin with CS n given by the equation u v + · · · + u n v n = 1 − z n +1 . For the right hand side we have the factor C − z n +1 = 1 and thefactor 2 for z n +1 = 1. The equation c = 1 − z n +1 = 0 gives the factor u v + · · · + u n v n = c, which is in fact CS n − while for c = 0 we have theequation u v + · · · + u n v n = 0 , which is N C n − = 1 + CS n − · ( C − . So in total we have CS n = CS n − · ( C −
2) + 2 + 2 CS n − · ( C − CS n − · ( C − − CS n − · ( C −
2) + 2 CS n − · ( C −
1) + 2= CS n − · ( C −
2) + CS n − · C + 2and, therefore, CS n = CS n + CS n − = CS n − · ( C −
1) + CS n − · C + 2 . xamples of Morphological Calculus 33Using the induction hypothesis CS n − = CP n − and C S n − = CP n − · S n − , this gives rise to CS n = C n − C n − + 1)( C n + C n − + · · · + C ) + 2= ( C n + C n − + · · · + C n ) + ( C n + C n − + · · · + C + 1)= ( C n + 1) CP n = CP n · S n . For the other case CS n +1 we note that CS n +1 is given by the equation u v + · · · + u n v n = 1 − u n +1 v n +1 giving the factor (Phantom projectiveplane) C − C + 1 for c = 1 − u n +1 v n +1 = 0 and C − u n +1 v n +1 = 1.Again for c = 0 we have the equation u v + · · · + u n v n = c = 0, leading tothe factor CS n − and for c = 0 we get factor N C n − as before. This leadsto CS n +1 = CS n − · ( C − C + 1) + (1 + CS n − · ( C − · ( C − CS n − · ( C − C + 1) − CS n − · C + C − , which, using the formulae for CS n − and C S n − yields CS n +1 = C n · ( C n +1 − S n +1 · R n · R + and so we finally get CS n +1 = CS n +1 + CS n = C n · ( C n +1 −
1) + ( C n + 1) ( C n + · · · + 1)= C n +1 + C n + · · · + C n − C n + C n + · · · + 1 = CP n +1 = S n +2 · CP n = S n +2 · S n +1 S . These calculations show a certain consistency in which the Poincar´e sphere S n and complex projective spaces CP n play a central role. Also the Phan-tom complex projective plane C − C + 1 reappears here as the set of points( u, v ) ∈ C which lie outside the hyperbola uv = 1; it is the new geometricinterpretation for strange phantom plane that arises from the morphologicalanalysis.Finally also the “bipolar plane” C − C − n , but in morphological calculus weare not interested in generality, only in canonical objects.Our next investigation concerns “Null Graßmannians”.By N G n,k ( C ) we denote the manifold of all k -dimensional subspaces of thenullcone N C n − in C n . Hence in particular N G n, ( C ) = C S n − . Let us makethe morphological analysis; once again there are two ways.Let V ⊂ N C n − be a k -dimensional complex subspace spanned by k -vectors τ , . . . , τ k . Which are of course linearly independent and satisfy: τ j = ( t j + i s j ) = 0 , i . e ., t j ⊥ s j & | t j | = | s j | h τ j , τ k i = h t j , t k i − h s j , s k i + i ( h t j , s k i + h t k , s j i ) = 0 . Next consider on C n the Hermitian inner product ( z, w ) = P nj =1 z j w j ;then we can normalize vector τ , i.e., ( τ , τ ) = h t , t i + h s , s i = 2, whichtogether with t ⊥ s | t | = | s | means that the pair ( t , s ) ∈ V n, ( R ) is themanifold of orthonormal 2-frames.Next one may choose τ such that ( τ , τ ) = 0 & | τ | = 2 with τ = t + i s . This automatically implies that h τ , τ i = h τ , τ i = 0 , i . e ., h t , τ i = h s , τ i = 0so that the pair ( t , s ) is an orthonormal 2-frame that is also orthogonal tospan R { t , s } , i.e., ( t , s , t , s ) ∈ V n, ( R ) . Continuing the reasoning, we may choose t j , s j in such a way that( t , s , t , s , . . . , t k , s k ) ∈ V n, k ( R ) = SO ( n ) SO ( n − k ) ;a necessary condition for this is n ≥ k .Now let ( τ ′ , . . . , τ ′ k ) be another k -tuple for whichspan { τ ′ , . . . , τ ′ k } = V & | τ ′ j | = 2 , ( τ ′ j , τ ′ k ) = 0 , j = k ;then there exists the unique matrix A ∈ U ( k ) such that τ ′ j = P kℓ =1 A jℓ τ ℓ .Hence we obtain the identity in terms of homogeneous spaces and in mor-phological sense N G n,k ( C ) = SO ( n ) U ( k ) × SO ( n − k ) = S n − · S n − · · · S n − k S k − · S k − · · · S . So, in the case n = 2 m is even, we have that N G n,k ( C ) = S m − · · · S m − k S k − · · · S = G m,k ( C ) · S m − · · · S m − k while for n = 2 m + 1, odd, we have N G n,k ( C ) = S m · S m − · · · S m − k +1 S k − · · · S = G m,k ( C ) · S m · · · S m − k +2 . This also implies that
N G m,k ( C ) is integrable.Another way of calculating the quantity makes use of the complex com-pact spheres CS n − that were obtained in terms of complex analysis. Usingthe notation N G n,k ( C ) for the corresponding null Graßmannians we have: N G n, ( C ) = CS n − ,N G n, ( C ) = { ( τ , τ ) ∈ CS n − · S × CS n − · S } mod U (2)= CS n − · S · CS n − · S S · S = C S n − · C S n − CP , xamples of Morphological Calculus 35and in general N G n,k ( C ) = C S n − · · · C S n − k CP k − · · · CP . Hence, in case n = 2 m we obtain N G m,k ( C ) = C S m − · · · C S m − k CP k − · · · CP = CP m − · S m − · · · CP m − k · S m − k CP k − · · · CP = CP m − · · · CP m − k CP k − · · · CP S m − · · · S m − k = G m,k ( C ) · S m − · · · S m − k and similarly for n = 2 m + 1 we get N G m +1 ,k ( C ) = G m,k ( C ) · S m · · · S m − k +2 , and so these objects are also integrable. The calculus of nullcones and thingscan also be done in real variables. Let R p,q be the space R p,q = R p + q withquadratic form | x | − | y | = p X j =1 x j − q X j =1 y j , ( x, y ) ∈ R p,q . Then the nullcone
N C p,q is the set of solutions ( x, y ) of equation | x | = | y | ;it contains of course (0 ,
0) and for | x | ∈ R + we have ( x, y ) = ρ ( ω, ν ) with ρ > x, y ) ∈ S p − × S q − . Hence, we have relation N C p,q = S p − · S q − · R + + 1 . By S p − ,q − we denote the set of 1 D subspaces of N C p,q ; it may berepresented by the equivalence classes ( ω, ν ) ∼ ( − ω, − ν ), ( ω, ν ) ∈ S p − × S q − . In morphological notation we have: S p − ,q − = N C p,q − R − S p − · S q − · R + R − S p − · S q − S p − · RP q − . For example for q = 2 we may put ν = (cos θ, sin θ ) and S p − , may beidentified with the equivalent pairs ( ω, cos θ, sin θ ) ∼ ( − ω, − cos θ, − sin θ ),which is equivalent with the Lie sphere S p − , ∼ = LS p = { e i θ ω : ω ∈ S p − , θ ∈ [0 , π [ } . But there is also another calculation of this manifold that leads to anotherquantity S p − ,q − and it corresponds to the “conformal compactification” R p − ,q − of R p − ,q − . To find this, let ( x, y ) = ( x ′ , x p ; y ′ , y q ) with ( x ′ , y ′ ) ∈ R p − ,q − . Then first we may intersect the nullcone with the plane x p − y q = 1,i.e., we put x p = 12 (1 − ρ ) , y q = −
12 (1 + ρ ) . | x | = | y | for the manifold S p − ,q − gives us ρ = | x ′ | −| y ′ | so that ( x ′ , y ′ ) ∈ R p − ,q − freely and then ( x p , y q ) are fixed. So thispart of S p − ,q − is equivalent to R p + q − . The remaining part of S p − ,q − isrepresented by the nonzero vectors λ ( x, y ), λ ∈ R \ { } for which x p = y q ;there are two cases: • If x p = y q = 0 we may normalize x p = y q = 1 and we have ( x, y ) =( x ′ , , y ′ ,
1) together with the equation | x ′ | − | y ′ | = 0. So this part of S p − ,q − is equivalent to the modified nullcone N C p − ,q − = 2 S p − ,q − · R + + 1 . • If x p = y q = 0 we have λ ( x ′ , , y ′ ,
0) with λ ∈ R \ { } and | x ′ | 6 = 0 and | x ′ | = | y ′ | , which is the definition of S p − ,q − .So the total morphological calculation becomes S p − ,q − = R p + q − + S p − ,q − (2 R + + 1) + 1 , = R p + q − + S p − ,q − · R + 1 , or, in terms of compactification of R p,q : R p,q = R p + q + R p − ,q − · R + 1 . case 1. : for q = 0 we simply obtain R p, = R p = R p + 1 = S p . case 2. : compactified Minkowski space-time R p, = R p +1 + R p − , · R + 1 , = R p +1 + ( R p − + 1) · R + 1 = ( R p + 1)( R + 1)= S p · RP . More in general we obtain for p ≥ q R p, = R p +2 + R p − , · R + 1 , = R p +2 + ( R p + R p − + R + 1) · R + 1 = ( R p + 1)( R + 1)= R p +2 + R p +1 + R p + R + R + 1= ( R p + 1)( R + R + 1) = S p · RP , and, continuing in this way we obtain for p ≥ q : R p,q = ( R p + 1)( R q + · · · + R + R + 1) = S p · RP q , as expected from the similar (but different) formula S p,q = S p · RP q . So once again we have two different quantities that are obtained in twodifferent canonical ways from what is mathematically considered to be onemanifold.Note that in particular (and this is weird) R m,m = ( R m + 1)( R m + · · · + R + 1)= R m + · · · + R m +1 + 2 R m + R m − + · · · + 1= ( R m + 1) + R · ( R m + 1)( R m − + · · · + 1) . xamples of Morphological Calculus 37For the classical Minkowski space time we get R , = ( R + 1)( R + 1) , and this is indeed projective line bundle over 3-sphere.Compactified complexified Minkowski space-time is given by CS = ( C + 1)( C + C + 1) = C + C + 2 C + C + 1= C + C · ( C + 1) + 1 = C + N C , with N C = C · CS + 1 , so it is not just replacing “ R ” by “ C ” in R , .We must still calculate the null Graßmannians N G p,q ; k ; they are definedas manifold of k -dimensional subspaces of the nullcone N C p,q = S p − · S q − · R + + 1 . Let V be such k -dimensional plane; then V is spanned by the basis ofthe form: e + ǫ , e + ǫ , . . . , e k + ǫ k ; e , . . . , e k ∈ S p − ; ǫ , . . . , ǫ k ∈ S q − ;orthonormal frames, so these bases belong to: e + ǫ ∈ S p − · S q − = 2 N C p,q − R − S p − ,q − , up to e k + ǫ k ∈ S p − k · S q − k = 2 S p − k,q − k , and within V the total quantity of such bases is given by O ( k ) = S k − · S k − · · · S . We thus have the morphological representation (with p ≥ q, q ≥ k ) N G p,q ; k = (cid:0) S p − ,q − (cid:1) · · · (cid:0) S p − k,q − k (cid:1) S k − · S k − · · · S = S p − ,q − · · · S p − k,q − k RP k − · · · RP = S p − · S p − · · · S p − k S k − · · · S S q − · · · S q − k = G p,k ( R ) · S q − · · · S q − k . Again there is another way of computing this whereby in the above, S p,q is replaced by S p,q = R p,q , leading up to the stereographic null-Graßmannian: N G p,q ; k = (cid:0) S p − ,q − (cid:1) · · · (cid:0) S p − k,q − k (cid:1) S k − · · · S = R p − ,q − · · · R p − k,q − k RP k − · · · RP = S p − · · · S p − k · RP q − · · · RP q − k RP k − · · · RP = G q,k ( R ) · S p − · · · S p − k . All these manifolds give hence rise to integrable morphological objects.For the Minkowski space-time we have the manifold of null-lines (lightrays):8 Frank Sommen N G , = R , · R , RP = ( R + 1)( R + 1)( R + 1) R + 1= ( R + 1)( R + 1) = S · S , i.e., the real twistor space.The last case we consider here is that of the space C p,q = C p + q providedwith the pseudo-Hermitian form(( z, u ) , ( z ′ , u ′ )) = ( z, z ′ ) − ( u, u ′ ) = p X j =1 z j z ′ j − q X j =1 u j u ′ j . The nullcone (( z, u ) , ( z, u )) = 0 is denoted by N C p,q ( C ) and it has realcodimension one, so its real dimension equals 2 p + 2 q − . The equation is | z | = | u | so: N C p,q ( C ) = S p − · S q − · R + + 1 . By T p,q we denote the manifold of one dimensional complex subspaces of N C p,q ( C ); it is a real submanifold of CP p + q − of real codimension one thathence subdivides CP p + q − in 3 parts and T , ⊂ CP corresponds to “realtwistor space” (the manifold of light-lines in Minkowski space). From thedefinition we have T p,q = N C p,q ( C ) − C − S p − · S q − · R + S · R + = S p − · CP q − , so in particular T , = S · CP = S · S . There is another approach leading to the twister space T p,q with a dif-ferent quantity.To that end we write ( z, u ) = ( z ′ , z p , u ′ , u p ) and consider the intersection N C p,q ( C ) ∩ { z p − u q = 1 } , which allows us to write z p = 12 (1 + ρ + i α ) , u q = 12 ( − ρ + i α ) . The equation for the point ( z, u ) now becomes | z ′ | − | u ′ | + 14 (cid:0) (1 + ρ ) + α (cid:1) − (cid:0) (1 − ρ ) + α (cid:1) = 0or ρ = | z ′ | − | u ′ | & α ∈ R . Hence the 1 D complex subspaces of N C p,q ( C ) that intersect the plane z p − u q = 1 are representable by vectors of the form (cid:0) z ′ , (1 + ρ + i α ) , u ′ , ( − ρ + i α ) (cid:1) with ( z ′ , u ′ ) ∈ C p − ,q − and α ∈ R . So this part of T p,q has quantity C p + q − · R . The other points of T p,q have the form ( z ′ , λ, u ′ , λ ) so there aretwo cases • λ = 0 , in this case we normalize λ = 1 and we have the equation | z ′ | − | u ′ | = 0 , giving a version of nullcone: N C p − ,q − ( C ) = T p − ,q − · ( C −
1) + 1xamples of Morphological Calculus 39 • In the case λ = 0 we have the point ( z ′ , , u ′ ,
0) with equation | z ′ | = | u ′ | and determined up to a constant c ∈ C \ { } , i.e., we get T p − ,q − .This leads to the recursion formula for T p,q with p ≥ q : T p,q = C p + q − · R + T p − ,q − · C + 1 . So in particular we get T p, = C p − · R + 1 = R p − + 1 = S p − , T p, = C p · R + (cid:0) C p − · R + 1 (cid:1) C + 1= (cid:0) C p + C p − (cid:1) R + C + 1 = (cid:0) C p − · R + 1 (cid:1) ( C + 1)= S p − · CP , T p, = C p +1 · R + (cid:0) C p − · R + C p − · R + C + 1 (cid:1) C + 1= (cid:0) C p − · R + 1 (cid:1) (cid:0) C + C + 1 (cid:1) = S p − · CP and so, continuing in this way, we obtain for p ≥ q : T p,q = (cid:0) C p − · R + 1 (cid:1) (cid:0) C q − + · · · + C + 1 (cid:1) = S p − · CP q − . In particular we re-obtain the expected formula T , = ( C · R + 1)( C + 1) = S · CP = S · S . The manifold CP is itself called the complex twistor space; it decomposesinto real twistor space T , together with two equals parts correspondingto | z | < | u | and | z | > | u | . In morphological language we have the cuttingexperiment CP − T , = C + C + C + 1 − ( CR + 1)( C + 1)= ( C − C · R )( C + 1) = C · ( C − R )( C + 1)= 2 C · C + · ( C + 1) , which indeed gives 2 copies of C · C + · ( C + 1) whereby we put C + = C − R = R · R − = R · R + . We can also calculate the null-Graßmannian
N G p,q ; k ( C ) of k -dimensionalcomplex subspaces of N C p,q ( C ). Let V be a complex k -subspace; then theframes ( t ; s ) , . . . , ( t k ; s k ) may be chosen such that ( t j ; t k ) − ( s j ; s k ) = 0,of course, but we may also choose ( t j ; t j ) to be the Hermitian orthonormalframe, i.e., ( t j ; t k ) + ( s j ; s k ) = 0 and | t j | = | s j | = 1.So in fact we can choose( t ; s ) ∈ S p − × S q − , ( t ; s ) ∈ S p − × S q − , and so on. Moreover these frames per plane V can be chosen in U ( k )-differentways, leading up to the morphological formula for p ≥ q ≥ k N G p,q ; k ( C ) = S p − · · · S p − k · S q − · · · S q − k S k − · · · S · S = S p − · · · S p − k · CP q − · · · CP q − k CP k − · · · CP . A similar calculation can be made using the stereographic spheres, leadingto: N G p,q ; k ( C ) = (cid:16) T p,q · ( C − R + (cid:17) · (cid:16) T p − ,q − · ( C − R + (cid:17) · · · (cid:16) T p − k +1 ,q − k +1 · ( C − R + (cid:17) U ( k )= S p − · · · S p − k · CP q − · · · CP q − k CP k − · · · CP , and in particular for p = q = 2, k = 2 we obtain: N G , ( C ) = S · S · CP CP = ( R + 1)( R + 1) , which corresponds to the real compactified Minkowski space.The compactified complex Minkowski space corresponds to: G , = S · S S · S = S · CP = ( C + 1)( C + C + 1) = CS , as can be shown using bivectors and Klein quadric.
6. Conclusions and Remarks (i).
COMPLETENESSMorphological calculus is best compared with a museum. It con-sists of a lots of special names, algebraic expressions and calculationsthat stand for geometrical objects and operations on these objects.In this paper we presented morphological calculus for the mostimportant classical manifolds. Like any museum, also our collection isincomplete. For example a full morphological treatment for the spingroups Spin( m ) and Spin( p, q ) is still to be done and there is a vastcollection of special manifolds or objects to be added to the catalogue.In building up our museum we give preference to the most interest-ing special manifolds (canonical manifolds) as well as to the “simplestways of introducing them”. So in fact the calculus is entirely based onexamples of objects and experiments; there is no idea of “a general man-ifold” and no theory behind the scene. (ii). CORRECTNESSMorphological calculus is correct in the sense that it takes spacewithin the language of calculus that is a correct language based on clearrules. This leads to the notion of quantity, which is in fact what a man-ifold becomes once it is introduced within the calculus language. Thisxamples of Morphological Calculus 41is practically done by assigning a name to an object along with an al-gebraic relation that expresses the definition of the object in calculus.The notion of quantity is somewhat comparable to the notions of cardi-nality and of volume that are used to express the contents or size of anobject. But there is no mathematical definition for it; it is an imaginarysubstance that resides entirely within the calculus.The main problem is not the calculus itself but the way of translat-ing objects of geometry into calculus expressions (morphological analy-sis); it usually happens that one and the same object can be translatedinto morphological language in many ways and that may cause confu-sion. To give an example, the compactified Minkowski space is given by R , = R + R ( R + 1) + 1 = ( R + 1)( R + 1)whereby R is the usual Minkowski space and R ( R + 1) + 1 = (2 R + S + 1) + S is a compactified light cone at infinity whereby we made use of stere-ographic sphere S . What would happen if we replace S by the usualsphere S Well, we would get in total: R + R (2 R + 2 R + 2) + 1 = R + 2 R + 2 R + 2 R + 1= ( R + R + R + 1)( R + 1) = RP · S S S S · S · S . This is no longer Minkowski space-time, yet there exists a meaningfulinterpretation for this object, namely the manifold of pairs ( e i θ ω, − e i θ ω )in C , with e i θ ω ∈ LS , the Lie sphere. For this manifold the above cal-culation makes sense. So the problem is not only to know what calcu-lation to make to describe an object correctly but also how to correctlyinterpret a calculation (morphological synthesis). It often happens thatdifferent objects turn out to share the same quantity.There is no way to avoid these problems; one simply has to ex-periment until one finds the best fitting calculations or interpretations.This may be seen as a drawback, but we see it as a stronghold thatillustrates the richness of the morphological language. (iii). CONSISTENCYMorphological calculus may be compared to making the bill of ameal in a restaurant; usually the bill adds up correctly but sometimesthe sum of the ingredients of the meal is more expensive than the meal.Here is a example in morphological calculus: Consider the space R n of bivectors in a Clifford algebra: b = P i
2. for r > r = 0 we get b = rω ∧ ν with r ∈ R + and ω ∧ ν ∈ g G , ( R ) = S · S , so in total S · S · R + = S · ( R − .
3. in case r = r = r > b = r ( ω ∧ ν + ω ∧ ν ) , whereby either ω ∧ ν = ± e ω ∧ ν , so b = rω ∧ ν (1 ± e ).Hereby ω ∧ ν may be chosen to belong to g G , ( R ) = S becausein fact every bivector b ∈ R may be decomposed uniquely intoself-dual and anti-self-dual parts: b = 12 (1 + e ) b + + 12 (1 − e ) b − , b ± ∈ R . So in the above case, ω ∧ ν ∈ g G , ( R ) is unique, so that the mor-phological contribution is given by2 R + · S = 2( R − . xamples of Morphological Calculus 43Hence, adding up (1) + (2) + (3), we get a total morphological sumof (2 R + + 1) S · ( R −
1) + 2( R − R · ( R + 1) + 2) · ( R − = ( R + 1)( R −
1) = R − . The gap in the calculation lies in the difference between R · ( R + 1) + 2and S = R + 1 . If in the above we would replace R + 1 = S by2 R + 2 R + 2 = S we would get a factor R · S + 2 = S and replacingthen S by S would make the bill add up correctly.So in fact R · ( R +1)+2 may be interpreted as an oversized versionof the Poincar´e sphere S = R + 1 . Also for the bivector space R we have three cases:1. in case r > r > R · S · S S · S · S S = ( R − · S · ( R − r > r = 0 we obtain: R + · S · S S = ( R − · S
3. in case r = r = r > r ( ω ∧ ν + ω ∧ ν ) in R ; the number of choices for span { ω , ν , ω , ν } equals g G , ( R ) = S while for each choice we have the quantity 2 R + S as before, leading to a total of S R + S ) = ( R − · S . So, the total bill for R reads( R − R ) R + 2 R + 2 R + 2)= ( R − R + ( R + 1)( R + R + 1) + 1) , while we would need the second factor to be equal to R + 1 tomake the bill add up correctly.From n ≥ R n is much more complicated sowe won’t do it here, but in any case we won’t get just R ( n ). This maybe seem as an inconsistency which is likely to repeat itself in cases ofpartitions of geometrical objects. We have no solution as even explana-tion of this, but it is clear that one can study this phenomena withinthe language of morphological calculus, which in itself is consistent. (iv). CALCULUS STYLESA calculus style is obtained by making certain restrictions on theuse of the calculus language and by a certain kind of application orfocus.In the canonical style we decided to replace the relation R = 2 R +1by its more rigorous form R = 2 R + + 1 in order to avoid too manyunwanted identifications.4 Frank SommenThis leads to the possibility to apply the rules of calculus on a freebasis (commutativity, brackets, etc.) whereby our focus is the calcula-tion of quantity for a large collection of manifolds and this calculationarises from a morphological analysis of the geometrical objects (andconstructions) we are interested in.In the formal style we start off from a given quantity, a polynomial a R n + · · · + a n with a > , a , . . . , a n ∈ N say, and we consider thecollection of all the algebraic expressions that evaluate to this quantity.Since we already start with a polynomial with positive integer coeffi-cients, we won’t consider any subtractions or divisions here, just addi-tion and multiplication. Also we won’t be using R + here and the relation R = 2 R + +1 will be replaced by a non-commutative and non-associativeversion of R = 2 R + 1 : R = R + 1 + R , R = R + ( R + 1) , the use of which leads to a change in the quantity. Also other calculationsinvolving commuting terms or factors or placing or removing bracketsare seen as as morphisms on the collection of morphological objects. Sofor each quantity we have basically a category.Parallel to this, for each polynomial a R n + · · · + a n we also havethe set of all geometrical objects that can be formed by glueing together(or not) a copies of R n , a copies of R n − , . . . , a n points. The focusnow is to study possible correlations between the category of algebraicexpressions and geometrical objects (graphs) for a given quantity; thisis morphological synthesis.For example for R + 1 we have two expressions R + 1 , R and two geometrical objects (apart from trivial disjoint union) semi-interval [0 ,
1[ or circle S and one possible correlation is to identify R + 1with [0 ,
1[ and 1 + R with the circle S .The more general case a R + b leads to a kind of calligraphy thatwe’ll study in forthcoming work. For this reason we will speak of calli-graphical calculus.
7. Outlook
It is not easy to provide complete references to the topic of morphologicalcalculus but certain examples of it as well as related topics are certainly avail-able throughout the mathematical literature. First of all there is our paper[6] in which we gave an introduction to morphological calculus which wassubdivided into an axiomatic approach, a canonical part and a formal partbased on the formal language of calculus. In this paper we focused mainlyon the canonical part by giving many more new examples of interesting cal-culations. Morphological calculus can be seen as a formal language and thetask of constructing good geometrical interpretations of calculations, calledxamples of Morphological Calculus 45morphological synthesis, can be seen as part of a research field called the the-ory of Lindenmayer systems (L-systems) for which there is a vast literature.We only refer to [5]. Also in the book [4] by Roger Penrose the language ofcalculus has been discussed, in particular the meaning of commutativity ofthe multiplication has been critically investigated. But the present paper ismostly concerned with examples concerning spheres, real and complex pro-jective spaces, special Lie groups and homogeneous spaces including Stiefelmanifolds and Graßmannians, various types of complex spheres and real andcomplex nullcones. All of this belongs to the theory of special manifolds (seee.g. [7]).In particular we also discussed real and complex compactified Minkowskispaces as well as twistor spaces which have many applications in mathemat-ical physics and for which we refer to the pioneering work [3] of R. Penroseand W. Rindler. Morphological calculus is of course also related to varioustopics in algebraic topology in particular Betti numbers, homology and co-homology, Poincar´e polynomials, Euler characteristics and much more thatis to be found all over the literature (use Wikipedia and see also [7]). Finally,many of our calculations also make use of bivector spaces, Clifford algebrasand Spin groups for which we refer to the books [1] and [2].
8. Acknowledgement
The author wishes to thank Dr. Narciso Gomes (University of Cape Verde- Uni-CV) for his help in the critical reading and the painstaking task oftypewriting this manuscript. We also wish to thank the referee for his valuablesuggestions during the preparation of the manuscript.
References [1] R. Delanghe, F. Sommen, V. Soucek, Clifford algebra and spinor valued func-tions, Mathematics and Its Applications 53, (Kluwer Acad. Publ., Dordrecht1992).[2] P. Lounesto, Clifford algebras and spinors (Second Edition), London Math.Soc. Lexture Note Series 286, (Cambridge University Press, Cambridge 2001).[3] R. Penrose, W. Rindler, Spinors and space-time, Volume 2, Spinor and twistormethods in space-time geometry, Cambridge Monographs on MathematicalPhysics, (Cambridge University Press, Cambridge, 1986).[4] R. Penrose, The emperors new mind: concerning computers, minds, and thelaws of physics, (Oxford University Press, Oxford, 1999).[5] G. Rozenberg, A. Salomaa, The mathematical theory of L-systems, (AcademicPress, New York, 1980).[6] F. Sommen, A morphological calculus for geometrical objects, J. Nat. Geom.15 (1-2), 1999, 1-64.[7] F. W. Warner, Foundations of differentiable manifolds and Lie groups,(Springer, New York, 1983).