Apollonian depth, spinors, and the super-Dedekind tessellation
AApollonian depth, spinors,and the super-Dedekind tessellation
Jerzy Kocik
Department of Mathematics, Southern Illinois University, Carbondale, [email protected] 8, 2020
Abstract
The configuration space of tricycles (triples of disks in contact) is shown to co-incide with the complex plane resulting as a projective space costructed fromthe tangency and Pauli spinors. Remarkably, the fractal of the depth functionsassumes a particularly simple and elegant form. Moreover, the factor spacedue to a certain symmetry group provides a parametrization of the Apolloniandisk packings.
Keywords:
Apollonian disk packing, Descartes theorem, depth function, Paulispinors, tangency spinors, modular group, experimental mathematics.
MSC:
1. Introduction
In the present paper we employ the concept of tangency spinors to parametrize thespace of tricycles. Quite surprisingly, it brings the depth fractal to a very regularform, that of the Apollonian belt. Additionally, in the same picture we obtain visu-alization of the classification of the Apollonian disk packing. The implied symme-try groups aextend the modular group and lead to a “super-Dedekind” tessellation,which extends the standard modular and Dedekind tessellations.We start with recalling a few concepts. The Apollonian depth function, intro-duced in [5], is defined as a function δ : R → N (with possible values 0 and ∞ ) asfollows: For a given triple of numbers ( a , b , c ), if any of them is non-positive, thevalue of δ is 0. Otherwise, one performs a process in R , one step of which consistsof replacing the greatest value in the triple by a + b + c − √ ab + bc + ca . (1.1)Repeat this step until the new number turns negative or 0. The number of stepsneeded to achieve it defines the value of δ ( a , b , c ). Here is an example for (179 , , , , π −−→ (62 , , π −−→ (23 , , π −−→ (3 , , −
1) (terminate)Thus δ (179 , , = a r X i v : . [ m a t h . M G ] S e p erzy Kocik Apollonian depth and spinors The geometric interpretation:
Any triple of mutually tangent disks (called furthera tricycle ) may be completed to an Apollonian disk packing. The depth functionis a measure how “deeply” is the given tricycle of curvatures ( a , b , c ) buried in thispacking. Every step of the process described above corresponds to replacing thesmallest disk by the greater from the two tangent to the triple ( a , b , c ). It is to be rununtil the external disk of negative (or zero) curvature is reached.Recall, that the curvatures of four mutually tangent disks satisfy the Descartesformula [3, 16, 1, 13, 10, 11]:( a + b + c + d ) = a + b + c + d ) (1.2)Its quadratic nature leads in general to two solutions: d = a + b + c ± √ ab + bc + ca , (1.3)which correspond to the two di ff erent disks that complete a given tricycle to aDescartes configuration, as illustrated in Figure 1.1. We choose the greater disk(smaller curvature) in defining the process while discarding the smallest disk (great-est curvature) from the original triple. This transformation will be called a Descartesascending move.
In general,
Descartes move is as above except the choice of thenew disk and the one to be discarted is arvbitrary. ab c ab c F igure a , b , and c . One of the solutions on the left side has negative curvature. The Apollonian depth is invariant under similarity transformations of the tricy-cles, i.e., under rotations, translations and dilations. In particular: δ ( a , b , c ) = δ ( λ a , λ b , λ c ) λ > a , b , c ) (cid:55)→ ( x , y ) = (cid:16) ac , bc (cid:17) (1.4)where we assumed that c = max( a , b , c ). The configuration space of the non-negative triples coincides with the unit square I . For economy, we shall use thesame symbol for this reduced depth function δ ( x , y ) = δ (1 , x , y ) erzy Kocik Apollonian depth and spinors δ obtained with a computer expriment. Thedegree of shade represents the value of the depth. The intriguing fractal-like patternresembles in parts that of the Apollonian disk packing, except the disks are replacedby ellipses.
1 1
44 4 4 44
44 4 F igure The fractal has a number of interesting properties, the most conspicuous beinga deformed Stern-Brocot structure in the pattern of the points of tangency. Theellipses in contact with the x axis are tangent to it at the squares of rational numbers. p / m . If fractions pm and qn satisfy pn − qm = ± x = ( p + q ) ( m + n ) . (1.5)A question arises: Is there a way to bring the ellipses to regular circles via somesimple transformation? “Unsquaring” the coordinates, suggested by the quadraticform of (1.5) is shown in Figure 1.3, left. Although interesting artistically, it didnot do the trick. Changing the plot to barycentric coordinates also fails, as shownFigure 1.3, right. In the present paper we employ the concept of tangency spinors,
33 3
33 33 3 33 33
444 4 44 444 44 4 4 444 4 4 4 44 44 4
55 5 F igure which, remarkably, results with a parametrization of tricycles via Argand plane, inwhich the depth fractal assumes a regular shape, that of the Apollonian Belt. Thesymmetries of the plane bring about a “super-Dedekind” tessellation, an extendedversion of the Dedekind tessellation. erzy Kocik Apollonian depth and spinors
2. Pauli spinors and tricycles
Recall that for a pair of two disks in contact in an Euclidean plane (the complexplane R (cid:27) C ) the tangency spinor is defined as a 2-vector or equivalently a com-plex number u = (cid:34) xy (cid:35) = x + iy = spin( A , B )such that u = wr r where w is a complex number representing the vector joining the centers of thedisks, and A = / r and B = / r denote both the circles, and their curvatures.Spinor is defined up to a sign. Recall that the arrows in the figures represent onlythe order of the disks (not the actual spinor). The remarkable properties of spinorsare presented in [12], and recapitulated in [7] (For the first appearance, see [9]) . C AB ab F igure Consider a tricycle with curvatures A , B , and C , and spinors a = spin( c , a ) and b = spin( C , B ), as in Figure 2.1. With the right choice of the signs of the spinors,we can write the following equations:( i ) a × b = C ( ii ) (cid:107) a (cid:107) = C + A ( iii ) (cid:107) b (cid:107) = C + B and ( iv ) (cid:107) a ± b (cid:107) = C + D ± ( v ) a · b = K (2.1)from which we will need initially the first three. The D ± stands for two curvatures ofthe two disks complementing ( A , B , C ) to the Descartes configuration, and K standsfor the curvature of the mid-circle that passes through the three points of tangencyof A , B , and C . (It is orthogonal to each of them.)We may combine the two spinors into a single tangency Pauli spinor , the vector ξ = (cid:34) ab (cid:35) ∈ C erzy Kocik Apollonian depth and spinors a and b understood as complex numbers (a vector which is much like thestandard Pauli spinor for describing the electron’s spin.) If the tricycle is consideredup to a scale and orientation, we may map ξ into a single complex number: ξ = (cid:34) ab (cid:35) −−→ (cid:34) b / a (cid:35) −−→ b / a = z = x + iy (2.2)Under this map, spinors a and b may be replaced by the following two (we keep thesame names a and b not to multiply symbols used): a = (cid:34) (cid:35) , b = (cid:34) xy (cid:35) (2.3)Using the associations (2.1), we get( i ) C = y ( ii ) A = − y ( iii ) B = x + y − y (2.4) Proposition 2.1.
The set of tricycles considered up to scaling and rotation may beparametrized by z = x + iy via Eq (2.4) , with the property δ ( z ) = δ ( A , B , C ) . (2.5)Now, we may execute a code for calculating the depth function as a function δ ( x , y ). The result is startling and is presented for positive x and y in Figure 2.2below. The color coding is from black to blue to read as the value of depth grows.The black regions correspond to zero depth. The bottom line of the colorful beltcoincides with the x -axis. The left side of the figure coincides with the y -axis. F igure ff erent depth are disks. We shall call the complex plane in this context the projective spinor space . Aspresented above, it is the result of projectivization of the tangency Pauli spinors: C × C ⊕ −−−→ C π −−−→ C P (cid:27) −−−→ ˙ C ≡ C ∪ {∞} erzy Kocik Apollonian depth and spinors F igure Let us start with a few basic observations: • Visually, the resulting fractal is similar to that of the Apollonian Belt (seeAppendix B). In that sense we get a surprisingly regular pattern, unlike thatof Figure 1.2 of [5]. • Regions of arbitrarily high values of the depth function exist in contact withthe regions of small values. Figure 2.3 shows a close-up of the corner regionnear the point (0,0). • The left-right symmetry is due to two mirror versions of regular tricycle (chi-ral versions). • The 0-depth regions correspond to the cases when one of the disks in a tricy-cle is of negative curvature, and may be derived from (2.4): • C < y < • A < y > • B < x + ( y − / < / • The pattern of the depth values in the Apollonian belt follows the order ofcompleting the initial three disks made by the 0-value regions, the disksbounded by circles y = y = x + ( y − / = / erzy Kocik Apollonian depth and spinors xy
01 12 200
22 22 33 33 44 44 F igure
3. Finite symmetries
The symmetries that transfer the Apollonian belt to itself and permute the regionsof depth 0 are easy to spot, see Figure 3.1. They are:(1) F : Reflection through the horizontal line y = / S : Inversion through the unit circle centered at origin, x + y = R : The Inversion through the unit circle centered at (0,1), x + ( y − = Proposition 3.1.
The above three transformations permute the values of curvaturesamong the three circles of the tricycles.F : C (cid:28) A S : A (cid:28) B R : B (cid:28) Cand preserve the patter of the depth function in Fig. 2.2.Proof.
The coordinate description ( x , y ) → ( x (cid:48) , y (cid:48) )above transformations are: F : (cid:40) x (cid:48) = xy (cid:48) = − y S : x (cid:48) = xx + y y (cid:48) = yx + y R : x (cid:48) = xx + ( y − y (cid:48) = x + y − yx + ( y − (3.1)As to the reflection F , the claim is obvious. For inversion S , calculate: C (cid:55)→ C (cid:48) = y (cid:48) = x + y − yx + ( y − = Cx + ( y − ∼ CA (cid:55)→ A (cid:48) = − y (cid:48) = − x + y − yx + ( y − = Bx + ( y − ∼ BB (cid:55)→ B (cid:48) = x (cid:48) + y (cid:48) − y (cid:48) = ... = Cx + ( y − ∼ C erzy Kocik Apollonian depth and spinors xy
01 12 200
22 22 33 33
RS FM F igure Thus all prove to be re-scaled by the same factor. For the third transformation, R ,similar calculations show that C (cid:48) = Cx + ( y − , A (cid:48) = Bx + ( y − , B (cid:48) = Cx + ( y − Thus the transformations F , S , R preserve the mutual ratios of the the curvatures,and consequently preserve the depth, hence the the conclusion of the invariance ofthe pattern. (cid:3) Yet another apparent feature of the image in Figure 2.4 is the left-right mirrorsymmetry:(4) H : Reflection through the vertical axis x = H : (cid:40) x (cid:48) = − xy (cid:48) = y (3.2)It does not a ff ect the Equations (2.4), but geometrically it represents the mirrorreflection of the tricycles.Denote the two finite groups generated by these transformations (with and with-out symmetry H ): Θ o = gen { S , F } Θ = gen { S , F , H } (3.3)Note that both groups contain also inversion R = S FS = FS F ∈ Θ o . The definingidentities are: H = S = F = R = ( S F ) = ( FR ) = ( RS ) = id . Moreover H commutes with all other elements. Group Θ has 12 elements and splits˙ C into 12 regions, each of which nay serve as the fundamental domain for group Θ , erzy Kocik Apollonian depth and spinors Q . The fundamental region for group Θ may be chosenas Q ∪ HQ .The essence of Proposition 3.1 may be now reformulated; Proposition 3.2.
The groups of symmetry (3.3) preserve the depth function: ∀ g ∈ Θ δ ( gz ) = δ ( z ) This results in characterization of the quotients of the action: Z / Θ o = (cid:26) all tricycles up to similarityexcluding reflections (cid:27) (cid:27) Q ∪ HQ Z / Θ = (cid:26) all tricycles up to similarityincluding reflections (cid:27) (cid:27) Q The latter, which disregards the chirality, coincides with the orbits of the al-gebraic definition of the process for triples as numbers. The fundamental regionrepresents all tricycles (up to permutations etc.). Note that every depth (color) isappears in these fundamental domains.The Möbius representation of the group elements will be given in the next sec-tion.
QSQ RSQFQSFQ RSFQHRSQHRSFQ HSQHQHFQHSFQ F igure C / Θ . Region Q is chosen as the fundamentaldomain. The x -axis passes through the center of the lower circle.erzy Kocik Apollonian depth and spinors
4. Apollonian packings and super-Dedekind tessellation
The projective space of the tangency Pauli spinors considered above may be alsoused to classify the Apollonian disk packing. This will lead us to yet another, infi-nite, partition of the plane, called below the super-Dedekind tessellation . Recallthat every tricycle ( A , B , C ) defines uniquely an Apollonian disk packing A ( A , B , C ).Define an equivalence relation which equates tricycles if they generate the sameApollonian disk packing (as usual, up to similarity):( A , B , C ) ∼ ( A (cid:48) , B (cid:48) , C (cid:48) ) if A ( A , B , C ) (cid:27) A ( A (cid:48) , B (cid:48) , C (cid:48) )We may execute this equivalence by adding yet another element of symmetry, namely T : z (cid:55)→ z + C AB ab a + ba − b F igure Here is why. Recall that the sum or di ff erence of the spinors a and b is the spinorfrom C to the disk that is inscribed between the two (see Figure 4.1). The two casescorrespond to the two ways of completing a tricycle to the Descartes configuration a = (cid:34) (cid:35) , b = (cid:34) xy (cid:35) ⇒ a = (cid:34) (cid:35) , b ± a = (cid:34) x ± y (cid:35) (4.1)In case of the spinors adjusted to the form (2.4), this corresponds to in the space ofspinors to shift to the left or right by one unit. Hence we define transformation: T : z (cid:55)→ z + H . Definition 4.1.
The
Apollonian symmetry groups are groups generated by thefollowing elements Ξ o = gen { T , S , F } Ξ = gen { T , S , F , H } (4.2) erzy Kocik Apollonian depth and spinors H in the group.)The tessellations due to the action of the group Ξ are now fragmenting the com-plex space into an infinite set of triangles and is shown in Figure 4.3. We shall callthis pattern the super-Dedekind tessellation for the reasons explained below.Analogously to Proposition 3.2, we have now Proposition 4.2.
The action of the groups (4.2) preserve the generated Apolloniandisk packings: ∀ g ∈ Ξ A ( gz ) = A ( z ) The corresponding quotients of the action are: Z / Ξ o = (cid:26) all Apollonian packings up to similarityexcluding reflections (cid:27) (cid:27) P ∪ HP Z / Ξ = (cid:26) all Apollonian packings up to similarityincluding reflections (cid:27) (cid:27)
Pwhere the fundamental domain for the action of Ξ may be chosen for instanceP = { ( x , y ) ∈ R | ≤ x ≤ / , y ≤ , and x + y ≥ } . Description
1. The super-Dedekind tessellation splits the plane C into triangles shown inFigure 4.3. It consists of triangles (60 ◦ , ◦ , ◦ ), each of which may serve as thefundamental domain of the action of group Ξ . The vertices have three types ofvalencies: 4, 6, ∞ The ones of valency ∞ are located along the circles of the Apol-lonian Belt drawn in red in Figure 4.3. The circles are those of the boundaries of theregions of constant depth δ of the depth fractal.2. The pints of the triangles are in 1-to-1 correspondence with all Apolloniandisk packings (up to similarity). The orbit of any of the points in it travels throughall tricycles contained in this packing (again, up to similarity).3. The super-Dedekind tessellation may be viewed as superimposing copies ofthe standard Dedekind tessellation. Indeed, the lower part in the plane, y <
0, aswell as the upper part, y >
1, both are congruent to the Dedekind tessellations ofPoincaré half-plane (see [14] and [6]. Note that the central disk of the underlyingApollonian Belt is tessellated by by the Dedekind tessellation of the unit disk. Infact, every red circle of A B contains an appropriate inversion of the Dedekind tes-sellation.4. If the mirror symmetry H is excluded from the group, the tessellation iscoarser and consists of triangles (60 ◦ , ◦ , ◦ ), made by gluing the former triangles.The upper and lower part of the tessellation will now correspond to the well-known erzy Kocik Apollonian depth and spinors Dedekind (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123) H SL ( Z ) (cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123) T S F S (cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)
F H (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)
12 regions (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) super − modular (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) super − Dedekind F igure tessellation of the Poincaré upper half-plane resulting by the action of the groupSL(2 , Z ), which in [6] is called modular tessellation.5. Finally, note that the upper (and the lower) part of the tessellation coincideswith the results of [7] where the integral packings were analyzed with the help ofthe tangency Pauli spinors.6. Algebraically, the super-Dedekind fractal is an extended version of the mod-ular and Dedekind tessellations. determined by the group SL(2 , C ). Similarly, thesuper-modular fractal is obtained. Both are obtained by adding an extra element tothe generators, see Figure 4.2. Remark:
The group Ξ has yet another meaning: it is a group of symmetries of theApollonian Belt. The basic element are:1. The inversions in 3-circles: circles that go through the points of contact ofany three disks forming a tricycle in the disk packing.2. The inversions in 2-circles: circles that are determined by any two circles incontact, namely as the circles that exchange them through the inversion. Theyalso pass through the contact points of the (infinite ) chain of circles inscribedbetween the two circles. Such a chain may be called a (generalized) Pappuschain .The proposed name “Apollonian symmetry group” is thus justified. The groupof concrete symmetries of other Apollonian packing is isomorphic to Ξ . This matterwill be discussed elsewhere. erzy Kocik Apollonian depth and spinors F igure A note on Möbius maps
The transformations (3.1) and (3.2) may be expressed in terms of linear fractionalmaps:
Geometricversion T : z (cid:55)→ z + = (cid:34) (cid:35) · zF : z (cid:55)→ ¯ z + i = (cid:34) i (cid:35) · ¯ zS : z (cid:55)→ − z = (cid:34) −
11 0 (cid:35) · ¯ zR : z (cid:55)→ i ¯ z ¯ z + i = (cid:34) − i (cid:35) · ¯ zH : z (cid:55)→ − ¯ z = (cid:34) i − i (cid:35) · ¯ z (4.3)Note that for consitency we should replace the translation T by reflection throughthe vertical line = /
2, i.e. by P : z (cid:55)→ − ¯ z = (cid:34) − i i i (cid:35) · ¯ z Translation is now recovered as T = PH , and T − = HP .Algebraic investigations may benefit from replacing the above with a subgroupof PSL(2 , Z [ i ]) with the following alternatives that omit the use of the conjugationof z : Algebraicversion ˆ T : z (cid:55)→ z + = (cid:34) (cid:35) · z ˆ F : z (cid:55)→ − z + i = (cid:34) i − i (cid:35) · z ˆ S : z (cid:55)→ − z = (cid:34) −
11 0 (cid:35) · z ˆ R : z (cid:55)→ izz − i = (cid:34) i − i (cid:35) · z (4.4)Note that the transformations ˆ T and ˆ S generate the modular group SL(2 , Z ). Weexcluded an equivalent of H , which would have to be ˆ Hz = − z , but this would allowfor FH ) n · z which moves z in the complex plane vertically, z (cid:55)→ z + ni , which cleaelydoes not belong to the symmetry groups of the pattern. erzy Kocik Apollonian depth and spinors
5. Addenda z The process π described in (1.1) for R may be redefined for the projective spinorspace ˆ C as a chain of maps that consistently push some initial z towards the “dark”region of δ =
0. We may facilitate it using two maps α, β : C → C . The first is theinverse of T : α : z → z − β is a combination of reflections S with conditional reflection F , and may bewritten as a single algebraic expression with the use of the absolute values, as shownin the box below.Initial value z can be chosen (or adjusted) to be on the right side of the plane,Re z >
0. The process is contained in the upper part of the color belt, 1 / ≥ y ≥ z to the “dark disk” | z − / | < /
4. Here is the algorithm spellout:
Algorithm redefined for z ∈ C
0. INPUT (x,y) x = | x | , d = ω = x + (1 / − y )
2. IF ω < /
4, RETURN d , ENDelseif x + ( y − < x → xx + ( y − y → + (cid:12)(cid:12)(cid:12)(cid:12) x + y − yx + ( y − − (cid:12)(cid:12)(cid:12)(cid:12) end if x : = | x − | d : = d +
13. GO TO 1.F igure
The string of operations is thus a word in the alphabet { α, β } , starting with α .Since β = id , the word is of the form α n βα n βα n B . . . βα n m for some m , and δ = (cid:88) i n i . erzy Kocik Apollonian depth and spinors C AB b b + aa + c c + F igure Two tangency spinors of Figure 4.1 totally determine the sizes of the tricycle andorientation in the Euclidean space (only the position is translationally not deter-mined). However, in trying to establish the map from the space of tricycles to thecouples of tangency spinors (or Pauli spinors):tricycles (up to translation) → C we encounter an ambiguity. The problem is in the choice of which disk is the anchorfort the spins a and b in Figure 4.1. We will tackle this ambiguity now. Figure 5.2shows a more regular notation for all possible spinors. With the notation shown wehave a = spin( B , C ) a + = spin( C , B ) = i ab = spin( C , A ) b + = spin( A , C ) = i bc = spin( A , B ) c + = spin( B , A ) = i ca + b + c = a + + b + + c + = b × a + = C , c × b + = A , a × c + = B . The complex representation is here more convenient because of the simple def-inition of the symplectic conjugation + . Recall that ( a + ) + = − a . Also, note thatchanging the sign in all spinors will not a ff ect the signs of the curvatures calculatedby the cross-products. The projection (2.2) should be now written as b , a + (cid:55)→ (cid:34) ba + (cid:35) (cid:55)→ z = ba + , and similarly for the other two pairs. We shall now show that the di ff erent choices ofanchors lead to complex number representation that di ff er by an equivalency defined erzy Kocik Apollonian depth and spinors Θ . z = ba + = zz = ac + = a − a + − b + = − i + ba + = − i − z = S F zz = cb + = − a − bb + = − a i b − b i b = i ab + i = i − − z = FS z = FS z
This defines a three-element group of Θ : G = { id , FS , S F } (cid:27) Z Note that we can use algebraic version of the transformations, i.e., FS = ˆ F ˆ S and S F = ˆ S ˆ F . (An even number of transformations in (4.3) will “cancel” the complexconjugations). The action of this group interchanges the regions X = { z (cid:12)(cid:12)(cid:12) Im z > / | and | | z | ≥ } Y = { z (cid:12)(cid:12)(cid:12) | z | ≤ | z − / | ≤ } Z = { z (cid:12)(cid:12)(cid:12) Im z > | and | z | ≤ / } , any of which can be chosen as the fundamental domain. Under the identificationdefined by the action of this group, we obtain the space topologically homeomor-phic to sphere, S , which is easy to see when we pick the cigar-like region Y andidentify the top and bottom edges. The whole spinor complex plane C ∪ {∞} , whichtopologically is also a sphere, “wraps” around the sphere Y three times, forming afiber bundle with discrete fibers coinciding with the orbits of G in ˙ C . erzy Kocik Apollonian depth and spinors Proposition 5.1.
Every rational disk packing is integral by scaling by some integer.Proof.
Pick a quadruple of disks in the packing in the Descartes configuration. As afinite set, it may be scaled to an integral quadruple. Hence all discs are integral. (cid:3)
Proposition 5.2.
Every rational Pauli spinor generates a (scaled) integral packing.Proof.
Equation (2.1) implies that the disk D that completes the tricycle determinedby z to the Descartes configuration is also rational. By Proposition 5.1, the claimholds. (cid:3) Proposition 5.3.
Only rational x,y generates (scaled) integral disk packing.Proof.
Referring to parametrization (2.4), since C is rational, so is y by (i). Since B = x + y − y is rational, so is x . Thus x = √ q for some q ∈ Q . Now, since D isrational, 1 + √ q ∈ Q (by (iv). Thus √ q ∈ Q , or q is a square. (cid:3) Hence, the integral packings correspond to (are coded by) the rational complexnumbers , z ∈ Q [ i ], forming a dense subset of the plane C . A related picture may befound in [7]. Appendix A: Apollonian disk packings An Apollonian disk packing is an arrangement of an infinite number of disks. Suchan arrangement may be constructed by starting with a tricycle, called in this contexta seed , and completing recursively every tricycle already constructed to a Descartesconfiguration.
66 66
12 1212 12 F igure Appendix B: The graph of the tricycles in an Apollonian pack-ing
The process that defines the value of the depth of a tricycle is just one of many pathsin the (discrete) space of tricycles in a given Apollonian disk packing. To see itin he context, let us define a graph in t: The vertices correspond to tricycles. Twotricycles are joined by an edge if they share two disks and the non-shared disks aretangent to each other in the packing. The process π makes a continuous (in the senseof the graph topology) path in this graph. Figure 5.4 presents this graph. As shown,It consist of an infinite number of tetrahedrons connected pairwise by the vertices.The vertices are 6-valent: this is obvious, for every tricycle may acquire anew diskin two ways, and lose one in three ways.b x x x x F igure The sub-graph of the tricycles that have value δ = F igure References [1] David W. Boyd, An algorithm for generating the sphere coordinates in a three-dimensional osculatory packing
Mathematics of Computation , , (122) 1973,pp. 369–377.[2] Dedekind, Richard (September 1877), "Schreiben an Herrn BorchardtÃijber die Theorie der elliptische Modul-Functionen", Crelle’s Journal, 83:265â ˘A ¸S292.[3] René Descartes, Oeuvres de Descartes, Correspondence IV, (C. Adam and P.Tannery, Eds.), Paris: Leopold Cerf 1901.[4] Lester R. Ford, Fractions, American Mathematical Monthly, (9) 45, 1938, pp.586â ˘A ¸S601.[5] Jerzy Kocik: Apollonian depth function and an accidental fractal,arXiv:2002.04135.[6] Jerzy Kocik: On the Dedekind tessellation, arXiv:1912.05768[7] Jerzy Kocik: Spinors, lattices, and the classification of the integral Apollonianpackings,. arXiv:2001.05866[8] Jerzy Kocik: A note on unbounded Apollonian disk packings,arXiv:1910.05924.[9] Jerzy Kocik: Cli ff ord Algebras and Euclid’s Parameterization ofPythagorean Triples, Advances in Appl. Cli ff . Alg. / kocik / papers / ff .pdf][10] Jerzy Kocik: A theorem on circle configurations, arXiv:0706.0372[11] Jerzy Kocik: Proof of Descartes circle formula and its generalization clarified,arXiv:1910.09174[12] Jerzy Kocik, Spinor structure of the Apollonian disk packing (submitted to Geometriae Dedicata )[13] Je ff rey C. Lagarias, Colin L. Mallows and Allan Wilks, Beyond the Descartescircle theorem, Amer. Math. Monthly
109 (2002), 338–361. [eprint: arXivmath.MG / (April 2008)[15] Processing.js, a software available at http://processingjs.org/ [16] Frideric Soddy, Kiss precise, Nature 137, 1021 (1936).https: // doi.org / //