AApollonian depth and the accidental fractal
Jerzy Kocik
Department of Mathematics, Southern Illinois University, Carbondale, [email protected] 12, 2020
Abstract
The depth function of three numbers representing curvatures of three mutu-ally tangent circles is introduced. Its 2D plot leads to a partition of the modulispace of the triples of mutually tangent circles / disks that is unexpectedly abeautiful fractal, the general form of which resembles that of an Apolloniandisk packing, except that it consists of ellipses instead of circles. Keywords:
Descartes theorem, Apollonian disk packing, depth function, el-lipses, Stern-Brocot tree, experimental mathematics.
MSC:
1. Introduction
The main purpose of this note is to present a fractal that results as a partition ofthe space of tricycles (i.e., three mutually tangent disks) into regions of constantvalues of the “depth function”. This function measures the depth a tricycle is buriedin the Apollonian disk packing that it determines. The structure of this fractal isunexpected, intriguing, and provides a rewarding object for further investigations.The fractal and most of its properties were discovered with the aid of computerexperimentation. We provide initial observations and a preliminary analysis of thefindings. One of the more intriguing outcomes is the occurrence of a deformedversion of Farey addition of proper fractions, Stern-Brocot tree, and a property ofellipses analogous to that of Ford circles.This is an example of visualization where geometry and number theory meet inan interesting way.
2. Basic notions
Any circle bounds two disks, the inner and the outer . The former is containedinside the circle, the latter extends outside of it and has infinite area in R . A disk hascurvature ± a if its bounding circle has radius 1 / a . The negative curvature is givento the outer disks. In the following, the disks are tangent if they are externallytangent , i.e., if they share only one point. A tricycle is a configuration of threemutually tangent disks (or circles). 1 a r X i v : . [ m a t h . M G ] F e b erzy Kocik Apollonian depth and the accidental fractal Descartes problem ”: given three mutually tangent circles,find the fourth that is simultaneously tangent to all of them. In the next letter heprovided the solution, known as the
Descartes formula , according to which thecurvatures a , b , c , d of four mutually tangent disks (now called Descartes configu-ration ), satisfy: ( a + b + c + d ) = a + b + c + d ) (2.1)The quadratic nature of Descartes’ formula assures in general two solutions: d = a + b + c ± √ ab + bc + ca (2.2)This is consistent with geometry: there are two di ff erent disks that complete a giventricycle to a Descartes configuration, as illustrated in Figure 2.1. ab c ab c F igure a , b , and c . One of the solutions on the left side has negative curvature. Another form of Equation (2.2) is a linear relation involving both solutions to Descartesproblem, say d and d (cid:48) : d + d (cid:48) = a + b + c (2.3)An Apollonian disk packing is an arrangement of an infinite number of disks.Two examples are presented in Figure 2.2. Such an arrangement may be constructedby starting with a tricycle, called in this context a seed , and completing recursivelyevery tricycle already constructed to a Descartes configuration.
66 66
12 1212 12 F igure major disk and its boundary the major circle . Remark:
A few clarifications concerning Figure 2.2 are in order. The greatest circlein the Apollonian Window (left) should be viewed as a boundary of an unboundeddisk extending outwards and having a negative radius and curvature, in this caseequal to ( − / circle of of the packing. Similarly, the ApollonianBelt is bounded by two lines that should be viewed as the boundaries of half-planes,understood as disks of zero curvatures. This convention allows one to see all pairsof tangent disks in Apollonian packings as tangent externally . These two examplesof packings are special: they have extra symmetries and the curvatures of all disksare integral (see labels inside the disks), hence the special names mentioned in thefigure’s caption. In the following we consider general cases.
3. Apollonian depth function
A tricycle determines an Apollonian packing uniquely. Also, any tricycle in a givenApollonian disk packing contained in it may serve as its seed. A natural questionarises:
The problem.
For a given tricycle, how many steps of inscribing new disks areneeded to reach the major circle of the Apollonian packing that it determines? Sucha number may be viewed as a degree of how deeply a particular triple is buried inthe network of the packing. It is sort of the “distance” of the original triple from theexternal disk and will be called the “Apollonian depth” of the tricycle. The goal isto visualize the topological space of tricycles, and the partition of this space definedby the depth function.The depth may be found by the following process: given a tricycle, form a newone by replacing the smallest circle by the greater circle of the two solutions to theDescartes problem (2.2). Repeat until you reach a disk of non-positive curvature.The number of steps of this process is the value we seek. We may give the processan algebraic form without reference to geometry:
Definition:
The
Apollonian depth is a function δ : R → N ∪ { , ∞} which takes the value zero if any of the three numbers is zero or negative. Otherwise,the value is determined by the dynamical process in R : T (cid:55)→ T (cid:55)→ T (cid:55)→ T (cid:55)→ ... (3.1)where T = ( a , b , c ) is the original triple and each new triple T n + is obtained by erzy Kocik Apollonian depth and the accidental fractal T n = ( a n , b n , c n ) by a n + b n + c n − (cid:112) a n b n + b n c n + c n a n (The minus sign of (2.2) to pick the greater disk, see .) The process is to be rununtil the first occurrence of zero or negative number in some T d . The number ofsteps d defines the Apollonian depth of the initial triple T .Here is an example: T = (15 , , (cid:55)→ (15 , , (cid:124) (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:125) T (cid:55)→ (15 , , (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) T (cid:55)→ (3 , , (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) T (cid:55)→ ( − , , (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) T Thus the depth of the triplet (15 , , δ ( T ) = Remark on unbounded packing:
Usually, the dynamical system (3.1) terminatesafter a finite number of steps. However, an infinite process is possible! Consider thefollowing triplet: T = ( ϕ − √ ϕ, , ϕ + √ ϕ ) (3.2)where ϕ = + √
52 is the golden ratio. Denote p = ϕ − √ ϕ and T = ( p − , , p ).The reader may check that the dynamical process in this case takes form:( p , , p − ) (cid:55)→ ( p , p , (cid:55)→ ( p , p , p ) (cid:55)→ ( p , p , p ) (cid:55)→ ... The process never ends with a negative curvature. In this case we define δ ( T ) = ∞ .For more on this unbounded arrangement of disks consult [4].
4. Visualization of the depth function
The Apollonian depth is invariant under similarity transformations of the tricycles,i.e., under rotations, translations and dilations. In particular: δ ( a , b , c ) = δ ( λ a , λ b , λ c ) λ > moduli space of tricycles up to this symmetry group. As such, it canbe parametrized by two numbers. One way is to scale tricycles so that its greatestcurvature becomes equal to 1. Thus the triple ( a , b , c ) is given coordinates( a , b , c ) (cid:55)→ ( x , y ) = (cid:16) ac , bc (cid:17) (4.1)where we assumed that c = max( a , b , c ). The moduli space T of the non-negativetriples coincides with the unit square T = I ⊂ R parametrized by ( x , y ) of (4.1),see Figure 4.1. There is an additional obvious redundancy, ( x , y ) ∼ ( y , x ), which willbe ignored for simplicity. erzy Kocik Apollonian depth and the accidental fractal
1 1 F igure For economy, we shall use the same symbol for this reduced depth function δ ( x , y ) = δ (1 , x , y )The plan is to visualize this function. We shall do it by associating to each point of T a color or shade representing the depth of the corresponding tricycle. F igure Result:
The image of this procedure is presented in Figure 4.2 The plot was ob-tained with the help of the program “processing.js” [6]. Appendix A shows the al-gorithm. The computing was done for 1000 × spiderweb ,or simply the web . The numbers in Figure 5.2 below indicate the depth values forselected plateaus.Besides the appealing image of a fractal, more secrets are brought with it, someof which are explored with the aid of computing. An alternative representation ofthe space of tricycles via barycentric coordinates is presented in Section 8.
5. Initial observations
Here we list some properties observed and verified experimentally.
1. The general pattern.
The spiderweb fractal consists of regions composedof points corresponding to tricycles of the same depth. The regions turn out to beellipses (as we will verify) except the the main large region in the right upper corner,which is parabolic. If presented in the standard 3D mode, the graph of the depthfunction δ : R → R has the shape of elliptic columns of di ff erent integral heights,quite like geological basalt fields, see Figure 5.1. Only several regions are includedto keep the image clear. One can see that the plateaus of drastically di ff erent heightsneighbor each other. F igure The web may be viewed as a packing of the square with ellipses. It resemblesan Apollonian disk packing. More accurately — it has the same tangency structureas the circle packing of a square presented in Figure 8.2, right. This is not exactly Apollonian packing since not all tangencies follow the Apollonian rule of erzy Kocik Apollonian depth and the accidental fractal
44 4 4 44
44 4 F igure The depth values in elliptic regions is shown in Figure 5.2. Note the tree struc-ture of the pattern presented in Figure 5.3 left. The vertices of this graph correspondto the ellipses, and the edges join regions that are tangent and di ff er in depth by 1.1234 F igure A few terms will be convenient: • The main x-wing chain is the sequence of the ellipses of depth 2,3,4,..., that aresimultaneously tangent to the parabola and the x -axis. (The corresponding disks inFigure 5.3 are made dark.) packing. The problem starts with the x-axis and the y-axis being mutually perpendicular, and propa-gates to the ellipses along the diagonal. However, each region enclosed by two consecutive ellipseson the diagonal and the x -axis (or y -axis) does follow the Apollonian rule of packing. erzy Kocik Apollonian depth and the accidental fractal • The parabolic main wing chain includes also the ellipses tangent simultaneouslyto the parabola and the y -axis. • The diagonal chain consists of ellipses simultaneously tangent to x -axis and the y -axis. Their depth values form a sequence 1, 2, 3, 4, 5... . • The corona of an ellipse (or parabola) in the web is the set of all web ellipsestangent to it. • The x -axis corona consists of all ellipses tangent to the x -axis. The y -axis corona is defined analogously. The parabolic corona consists of the ellipse tangent to theparabolic region of depth 1.
2. Warm-up, the first findings.
Inspection of the x -wing chain (regions of depth2,3,4,... etc) suggest that the x -coordinate of their points on the x axis seem to followthis simple pattern:13 , , , , ... and in general n − n + y -values of these points seem to follow also a simple rule:14 , , , , and in general 12 n ( n + (cid:32) n − n + , n ( n + (cid:33) (5.1) F igure An analogous experimental work suggests that the sequence of points separatingthe ellipses along the diagonal of the square T are rational (cid:32) n ( n + , n ( n + (cid:33) (5.2) erzy Kocik Apollonian depth and the accidental fractal • Yet another puzzling property is visible: each of the ellipses in the central x-chainseem to be tangent to the x-axis and the parabola at the same x-coordinate. • A property that is harder to notice is that all ellipses in the x -axis corona aretangent to the axis rational numbers squared. This will lead to a “squared” variationon the Ford fractions.
3. Basalt rock discontinuities
Intuition would suggest that a su ffi ciently smallchange of size of one of the circles in a tricycle can lead to a change in depth notgreater than 1, if any. It is not so: one of the conspicuous outcomes is that regionsof arbitrarily big jumps of depth may neighbor each other. For instance, consider avertical line through the point separating regions of depth δ = δ =
3, namely (9 / , / / , / + ε )It has depth 1 for ε =
0, and 3 for arbitrarily small positive values of ε , as is easy tofind out with the help of math software. The parametrized line (9 / , t ) splits into[0 , = { } (cid:124)(cid:123)(cid:122)(cid:125) depth = ∪ ( 0 , / (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) depth = ∪ [ 1 / , (cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) depth = Figure 5.5 illustrates such tricycle. Were the disk b = /
16 slightly bigger, the line(half-plane) underneath would become a large disk and the dynamical system wouldfirst acquire a disk to the right of a and b before becoming encircleable, hence thedepth of the tricycle is 3. b a1 F igure , / , / ∼ (9 , ,
1) lies on the boundary between depth 1 anddepth 3. In the chart, it corresponds to the point (1 / , / Clearly, one easily finds other points where the tricycle goes through arbitrarilygreat jumps in the value of the depth function δ . The cylinders of arbitrarily di ff erentheights can be mutually tangent in Figure 5.1. To exemplify it, a dense sample of avertical line at x = / y changing from 0 to the edge of parabola. Figure 5.6presents this cut of the graph of δ , with logarithmic scale in the value axis. erzy Kocik Apollonian depth and the accidental fractal e F igure , − ( , − √ ). Itconsists of 1000 sampling points. The vertical axis is in logarithmic scale Ln( δ )Note the drop at x = /
12 to the value of 2. on the vertical axis.
Here are a few selected points: (cid:32) , , (cid:33) → δ = (cid:32) , , + . (cid:33) → δ = (cid:32) , , + . (cid:33) → δ = (cid:32) , , + . (cid:33) → δ = (cid:32) , , + . (cid:33) → δ = The golden point (remark):
The red circle in Figure 5.4 encircles the point ofthe unbounded disk arrangement (3.2), for which the depth is equal to infinity. Itscoordinates are: (cid:16) ϕ − √ ϕ, (cid:0) ϕ − √ ϕ (cid:1) (cid:17) ≈ ( 0 . , . erzy Kocik Apollonian depth and the accidental fractal F igure Additional experimentation suggests that points of tangency of two regions oftwo di ff erent depths belong to the region of the smaller depth. Thus the regions arein general neither closed nor open.
6. Calculating the quadratic equations of the plateaus
Here we derive explicitly the equations for some of the elliptic regions in the weband show a general method for such calculations.
A. The parabolic plateau of depth 1.
Referring to Figure 6.1(left), the tricycle(1 , a , b ) is visibly of depth 1, but is in the state of a tipping point. Indeed, makingcircles a or b slightly bigger would turn the dotted line into a disk, increasing thevalue of the depth of the tricycle (1 , a , b ). Using (2.1), we can write this conditionas ( a + b + ≥ a + b + ) . With a little algebra, this can be rewritten (for the equal sign) as2( a + b ) = ( a − b ) + , (6.1)which is evidently a parabolic equation. Replace x = a − b , y = a + b to get2 y = x +
1. The main axis of the parabola coincides with the diagonal of the chart.Here is a recreational problem:
Problem:
What is the probability that three random circles can be encompassed by a circlewhen put in a mutually tangent position?
Solution:
Clearly, the answer depends on the choice of measure one imposes on the set ofcircles. Let us chose the uniform measure on the parameter of curvature. Using the aboveresult, via a simple integral one finds the area of the parabolic region to make 3 / P = . a b1 a b s F igure δ =
1, Right: δ = B. The elliptic plateau of depth 2.
We shall show that this region has an ellipticshape. Figure 6.1(right) shows a configuration in which disks (1 , a , b ) are separatedfrom the outer disk of curvature 0 (dotted line) by one intermediate disk of curvature s , hence it is of depth 2. The depth value is unstable: were the disk s slightly smaller,the dotted line would become an inner disk, and the depth of the configuration wouldincrease. Thus the boundary of the region of depth 2 corresponds to this type ofconfiguration. The implied equations are (cf., (2.2)): s = a + b + + √ abs = a + b + − √ ab + a + b Eliminating s , we get √ ab + √ ab + a + b = , which after being squared twice, leads to( a + b ) − ( a + b ) − ab + = x = a − b and y = a + b gives the standard form:12 x + (cid:16) y − (cid:17) = (cid:16) a + b − (cid:17) + a − b ) = C. The plateaus of depth 4.
Starting with he previous disk arrangements, we mayincrease the number of the disks separating the disk 1 from the “tipping’ dotted line.We shall see two examples of such arrangements for regions of δ =
4, see Figure6.2.
Case 1:
The idea is to write one quadratic equation, namely the Descartes formula,for a chosen Descartes configuration in this system, and write a set of linearized erzy Kocik Apollonian depth and the accidental fractal A ) (0 + s + x + y ) = + s + x + y )( B ) + s = x + y + s ) s + s = x + y + s ) s + = x + y + s )Part (B) consists of 3 linear equations, su ffi cient to express every s i in terms of x and y . Here we need s = / − x − y . Under substitution, (A) becomes a quadraticequation in x and y : 16 x + xy + y − x − y + = , which describes one of the ellipses bounding region od δ =
4, namely the one in thediagonal chain (labeled later as G , ). Case 1.5:
The quadratic equation for the system of equations may be chosen alsofrom a di ff erent Descartes configuration in the arrangement. For instance in theabove example we may consider the configuration involving disks x , y , and 1:( A ) (1 + s + x + y ) = + s + x + y )The 3 linear equations (B) stay the same, but now we need to extract a di ff erentunknown: s = / − x − y . Under substitution, the quadratic equation becomes:16 x + xy + y − x − y + = Case 2:
Similarly, for the third arrangement in Figure 6.2, we have:( A ) (1 + x + y + s ) = + x + y + s )( B ) + s = s + s + y ) s + x = s + s + y ) s + = x + y + s )Substituting s = / − x / − y / x + xy + y − x − y + = D. The general method.
To obtain other equations, one needs to consider thevariety of possible disk pyramidal arrangements, by which we mean the following:Set a line (disk of zero curvature) at the bottom. Set two disks mutually tangent,call it A and B. In the ideal triangle formed in the space enclosed, inscribe a systemof disks so that ... The smallest is of curvature 1. Choose two of the disks tangentto 1 and denote curvatures x and y . Denote the remaining curvatures by s , s ,..., s n .Now write a system of equations: (A) a quadratic equation for one of the Descartesconfigurations, and a system of linear equations (B) for the remaining chain of disksusing (2.3). erzy Kocik Apollonian depth and the accidental fractal x y x y 0 0 S S S S S S x y x y F igure δ =
4. Disks in the arrangements are deformed inorder to improve visualization. Each arrangement is accompanied by a graph:black vertex denotes the disk of curvature 1, the square vertex stands for thestraight line.
E. Labeling system
There is a problem of uniquely labeling the ellipses in thespiderweb. One way is to follow the ternary tree structure. E.g., counting from say2, 2DRRL means: move from 2 first down one step, two steps to the right and onestep to the left, every time decreasing the depth by 1, where L, R, D need to bedefined e.g, as “left”, “right”, and “down” with respect to the direction from the lastentry.Instead, we use the following labeling system: Deform the web so that the el-lipses become triangles organized in an orderly fashion, as shown in Figure 6.3.Note the benefits:1. Regions of the same value of the Apollonian depth function δ become con-gruent triangles.2. For a fixed value of δ the corresponding triangles form a matrix-like pattern.This allows one to use labeling: dL i , j where “L” is just a standard indication that this particular labeling is used, “ d ” isthe value of the depth, and i , j are the “coordinates” of the triangle within this setof triangles: i stands for the number of the column (from the left). and j stands forthe row (starting at the bottom). Figure 6.3 shows the labeling with space-savingomission of the letter “L”.Note that topologically the pattern is that of Sierpi´nski triangle. erzy Kocik Apollonian depth and the accidental fractal F. Quadratic equations.
Here are the equations for the first few plateaus, obtainedby the method outlined above.1 L , : x − xy + y − x − y + = L , : 4 x + xy + y − x − y + / = L , : 9 x + xy + y − x − y + = L , : 94 x + xy + y − x − y + = L , : 16 x + xy + y − x − y + = L , : 19681 x + xy + y − x − y + =
0A few curious things become at once visible.1. Often, but not always, the free term is reciprocal to the coe ffi cient at x . Insuch a case, the free term is the x-coordinate of the point the ellipse touchesthe x -axis.2. If the ellipse is tangent to the x -axis, its coe ffi cient at the term x is equal to 2.3. Another intriguing property is the following: completing the square leavesout the same term, 2 xy in each of the equations.The above equations may be thus rewritten with the squares completed and in aform that should make inspection easier:1 L , : ( x + y ) + = xy + x + y L , : (2 x + y ) + = xy + x + y L , : (3 x + y ) + = xy + x + y L , : (cid:32) x + y (cid:33) + = xy + x + y L , : (4 x + y ) + = xy + x + y L , : (cid:32) x + y (cid:33) + = xy + x + y A longer list of the ellipse equations may be found in Appendix B.The above properties suggested the theorem that is spelled out in the next sec-tion. erzy Kocik Apollonian depth and the accidental fractal L L , L L L L L L L L L L L L
14 1101 19 425116 49925 916 F igure , ≡ L , . In this graphic representation, the size of a trianglecorresponds to the value of the depth function.erzy Kocik Apollonian depth and the accidental fractal
7. Stern-Brocot structure
The most intriguing property of the fractal is appearance of Stern-Brocot stucture inthe pattern of the points of tangency.
Inspect the x -wing main chain, the chain of ellipses of depth 2,3,4, etc., that aresimultaneously tangent to the parabolic region and the x -axis. Using the equationsof the previous section we may find that the x -coordinates of the points they touchthe axis form the following progression of fractions:14 , , , , ..., ( n − n
16 1,25 25
25 4,49 49
25 9,64 64
49 91625492564 [1,1] [0,1] [1,0] [1,2] [2,1] [3,1] [3,2] [2,3] [1,3] F igure
14 14 F igure Further inspection reveals that the ellipses inscribed between them touch the x -axis at points that result via a deformed version of the Farey addition of fractions.Namely, the point for the inscribed ellipse between ab and cd is ab (cid:1) cd = ( √ a + √ c ) ( √ b + √ d ) (7.1)Iterating this process will account for all ellipses tangent to the x -axis, i.e., the x -corona. Figure 7.2 shows an order of recovering the points, which starts with theextreme fractions 0 / /
1, and then follows the deformed Farey addition.Before we collect these facts in one extended statement, let us recall the basicfacts of a Stern-Brocot tree.
Start with a 2-element sequence of pairs: [1,0], [0,1] (vectors). Create an array of se-quences, making new sequences from the previous by inscribing new terms betweenthe existing terms. The new terms are simply the vector sums of the neighbors: ..., [ a , b ] , [ c , d ] , ... (cid:55)→ ..., [ a , b ] , [ a + c , b + d ] , [ c , d ] , ... (7.2)The result should be called the Stern-Brocot array , the initial fragment is presentedbelow:[1 ,
0] [0 , ,
0] [1 ,
1] [0 , ,
0] [2 ,
1] [1 ,
1] [1 ,
2] [0 , ,
0] [3 ,
1] [2 ,
1] [3 ,
2] [1 ,
1] [2 ,
3] [1 ,
2] [1 ,
3] [0 , erzy Kocik Apollonian depth and the accidental fractal tree :[1 , , , ,
4] [4 ,
3] [3 , ,
5] [5 ,
2] [2 , , ,
5] [5 ,
2] [3 , ,
4] [4 , p , q ] (cid:55)→ pq , the tree becomes a tree of all positive rational numbers, and this is how the originaltree is usually defined and presented. In such a case, the the rule (7.2) is replacedby the so-called Farey addition of fractions: pq ⊕ st = p + sq + t erzy Kocik Apollonian depth and the accidental fractal The following property is analogous to the well-known Ford’s theorem for circles[3].
925 49 9162549 A: B:
33 1 3 1, ,88 88 8 88 = C: F: ,14 F igure Theorem 1:
For every reduced rational number p m in the closed interval [0 , ⊂ R draw an ellipse E [ p , m ] defined by (cid:32) mp x + p + m − pm y (cid:33) + p m = xy + x + m − p + m y (7.3) Then the following hold: (A)
Each ellipse lies above the x -axis and is tangent to it at the point p / m . (B) The interiors of the ellipses are disjoint. Moreover, if fractions pm and qn satisfy det (cid:34) p qm n (cid:35) = ± then there is an ellipse inscribed between these ellipses and the x -axis, namelyellipse E [ p + q , m + n ] over the point x = ( p + q ) ( m + n ) . (7.5) erzy Kocik Apollonian depth and the accidental fractal (C) Two ellipses satisfying (7.4) are mutually tangent at point: E [ p , m ] ∩ E [ q , n ] = (cid:32) p + q − m + n − , m + n − (cid:33) . (7.6) (D) The ellipses coincide with the plateaus of the constant values of the depthfunction δ . In particular, the value of δ in E [ p , n ] is equal to the row number of ( p , n ) in the Stern-Brocot tree X. Proof:
Lengthy calculations with a support of computer simulations.. (cid:3) pq st ( )( ) p sq t ++ x pq st p sq t q t + − + − + − F igure Special case 1:
Note: the main chain of ellipses is formed above the fractions suchthat m = p +
1. The equations of the ellipses have a detectable pattern (cid:32) p + p x + y (cid:33) + p ( p + = xy + x + p + y Special case 2:
The parabola is included in the x -corona, namely as E [1 ,
1] tangentto x -axis at x = (1 / =
1. Indeed, the general equation (7.3) reduces to theparabola equation 2( x + y ) = ( x − y ) + p = m = Special case 3:
The vertical line at x = E [0 , p = m = x = p before thesubstitutions). The theorem holds with the inclusion of this line. For instance, thepoints of the tangency of the the ellipses of the diagonal chain with the y -axis agreewith Equation (7.6) and are: E [1 , m ] ∩ E [0 , = (cid:32) , m (cid:33) : (cid:32) , (cid:33) , (cid:32) , (cid:33) , (cid:32) , (cid:33) , etc . erzy Kocik Apollonian depth and the accidental fractal Special case 4:
Each ellipse in the main x -wing chain is tangent to the x -axis andto the parabola at points that are vertically aligned. The coordinates on the parabolaare E [ p , p + ∩ E [1 , = (cid:32) p ( p + , p + (cid:33) . The statement (7.5) implies the following structure of the tangency points forthe x -corona: The parabolic corona consists of the ellipses tangent to the parabolic region of depth δ =
1. It includes the x-wing main chain, the y -wing main chain, and the ellipsesinscribed in the regions between them and the parabola. It turns out that the patternof the tangency follows the same Stern-Brocot structure with the deformed Fareyaddition, but now it applies to both coordinates x and y , and extends over the wholelength of the parabola. Proposition
For any positive value of p and q and n = p + q , the numbers a = p n , b = q n satisfy the parabolic Equation (6.1) , thus the points ( a , b ) lie on the parabola P . In this context, the pairs [ p , q ] will label both the points on the parabola withcoordinates Z (cid:51) [ p , q ] (cid:55)→ ( x , y ) = (cid:32) p n , q n (cid:33) ∈ R where n = p + q and ellipses tangent to the parabola at that points. Such ellipses will be denoted by F [ p , q ].The parabolic corona undergoes a phenomenon analogous to that of the x -axiscorona: erzy Kocik Apollonian depth and the accidental fractal Proposition:
The ellipses in the parabolic corona in the web are tangent to theparabola in the rational points of form ( x , y ) = (cid:32) p n , q n (cid:33) , n = p + q . (A) For every ( p , q ) ∈ N there is such an ellipse F [ p , q ] tangent to the parabolaat the above point. Additionally, two ellipses, F [ p , q ] and F [ p (cid:48) , q (cid:48) ] are mutuallytangent iff det (cid:34) p p (cid:48) m m (cid:48) (cid:35) = ± and det (cid:34) q q (cid:48) n n (cid:48) (cid:35) = ± m = p + q , m (cid:48) = p (cid:48) + q (cid:48) . (The two conditions are equivalent due to (6.1) .) The point of tangency is (cid:32) p + p (cid:48) − m + n − , q + q (cid:48) − m + n − (cid:33) (B) The ellipse inscribed between the above ellipses is F [ p + p (cid:48) , q + q (cid:48) ] , touching the parabola at the point with coordinates resulting from the deformedFarey addition: (cid:32) p m , q m (cid:33) (cid:1) (cid:32) p (cid:48) m (cid:48) , q (cid:48) m (cid:48) (cid:33) = (cid:32) ( p + p (cid:48) ) ( m + m (cid:48) ) , ( q + q (cid:48) ) ( m + m (cid:48) ) (cid:33) Figure 7.5 summarizes the main points. , p qn n , s tm m ( ) ( ),( ) ( ) p s q tn m n m + + + + p s q tn m n m + − + − + − + − , p qn n , s tm m F igure Note that as before, one may organize all ellipses in the parabolic corona andthe corresponding points on the parabola in a form of a tree. erzy Kocik Apollonian depth and the accidental fractal (cid:16) , (cid:17)(cid:16) , (cid:17)(cid:16) , (cid:17)(cid:16) , (cid:17) (cid:16) , (cid:17) (cid:16) , (cid:17)(cid:16) , (cid:17) (cid:16) , (cid:17) (cid:16) , (cid:17)(cid:16) , (cid:17)(cid:16) , (cid:17) (cid:16) , (cid:17) (cid:16) , (cid:17)(cid:16) , (cid:17) (cid:16) , (cid:17) For instance (cid:32) , (cid:33) (cid:1) (cid:32) , (cid:33) = (cid:32) (cid:1) , (cid:1) (cid:33) = (cid:32) (1 + (2 + , (1 + (2 + (cid:33) = (cid:32) , (cid:33) Figure 7.6 shows the first few steps of such recurrence. Note that the y-axis andthe x-axis are among the ellipses as the special cases F [0 , F [1 , [0,1] [1,0] [1,0] [1,1] [0,1] [1,0] [1,1] [0,1] [1,0] [1,1] [0,1] [1,2] [2,1] [1,2] [2,1] [1,3] [3,1] [2,3] [3,2] F igure Pseudoproofs:
The claims were tested by a computer process of evaluating thedepth function at the neighborhoods of the tangency points by probing such as δ ( a , b ) and δ ( a , b + ε )With Maple, the value of ε = . Remark:
The base points of the lower corona and the ceiling points of the uppercorona are vertically aligned, i.e., they share the same x -coordinate. erzy Kocik Apollonian depth and the accidental fractal
8. Barycentric coordinates
As mentioned in Section 4, the chart of the depth function may be represented inbarycentric coordinates. We denote them with double brackets. The curvatures of atricycle ( a , b , c ) of disks are rescaled to(( x , y , z )) = (cid:32) aa + b + c , ba + b + c , ca + b + c (cid:33) ( x + y + z =
33 33 3 33 33
44 4 4 44 444 44 4 4 444 4 4 4 44 44 4
55 5 F igure In this setup the three disks of the tricycles are distinguishable. The three verticesof the triangle correspond to two lines separated by a circle, each time a di ff erentdistribution: (1 , , , (0 , , , (0 , , / , / , / (cid:16)(cid:16) , , (cid:17)(cid:17) .This form of the Apollonian spiderweb is more symmetric and elegant, but for ex-perimentation, the rectangular framework of the previous sections is easier to ex-plore. erzy Kocik Apollonian depth and the accidental fractal F igure The rational points of the original spiderweb remain rational in barycentriccoordinates. E.g., the tangency points between ellipses in the main chain are now (cid:32)(cid:32) n + , n ( n − n + , n ( n + n + (cid:33)(cid:33) (and the permutations). Points of tangency between the region of depth 1 and el-lipses in the chain become: (cid:32)(cid:32) n − n + , ( n − n − n + , n n − n + (cid:33)(cid:33) Finally, the points of tangency between the diagonal ellipses are (cid:32)(cid:32) n ( n − n − n + , / n − n + , / n − n + (cid:33)(cid:33) erzy Kocik Apollonian depth and the accidental fractal
9. Additional bits
A. Apollonian disk packings in the chart.
Let ˆ p denote the Apollonian circlepacking (up to similarity) determined by tricycle p ∈ T . Define the equivalencerelation p ∼ q if ˆ p = ˆ q The quotient T / ∼ is the moduli space of Apollonian disk packings. Each individ-ual Apollonian disk packing understood as the set of all tricycles it contains may bedrawn as a dust of points in T . Figure 9.1 shows the equivalence class of tricyclescorresponding to the Apollonian Window. Since every point of T leads to an Apol-lonian packing, the space splits into an uncountable number of sets of countablemany points.Among problems that are interesting and easy to state but not necessarily simpleIs: Is there a continuous path in T that includes all Apollonian packings withoutrepetitions? F igure B. Size function.
A scalar function related to that of depth is a function f : T → R ∪ {∞} associating to every tricycle the radius (or curvature) of the size of the greatest circleof the Apollonian disk packing it determines. A version for the barycentric setup of erzy Kocik Apollonian depth and the accidental fractal g = sin ◦ f . Thefine pattern in some regions results from the interaction between resolution of thedrawing with the resolution of the rapidly changing values of g . F igure C. Squaring
Since the points of tangency are all squares of rational numbers, onemight think that re-scaling the figure so that the squares are brought to non-squareswill “straighten” the figure and the ellipses will become circles. Modifying the codeand rerunning the program invalidates the guess. But a dramatic image it producesis shown in Figure 9.3. F igure E. Open questions and challenges.1.
Find the exact equations of all ellipses in the fractal and a consistent way topresent them as function of the ellipses addresses (see Figure 6.3). The rationality of tangency points prompt further investigations into plausiblenumber-theoretic relations. Is there a single (conformal?) map that will transform all the ellipses of thespiderweb into circles?
Appendix A: The algorithm
The pseudo-code of the algorithm is presented below: for(int n = 0; n < 1000; n++){ for(int m=0; m <= n; m++){ new = 1; depth=0;T[0] = n; T[1] = m, T[2] = 1000;while ( (new > 0) && (depth < 21) ){ depth=depth+1;T = sort(T);float a = T[0]; float b = T[1]; float c = T[2];new = a + b + c - 2*sqrt(a*b + b*c + c*a);T[2] = new;} //end whilecolorRGB( 30*depth, 30*depth, 30*depth );draw point(n,m); draw point(m,n);} // end of ‘‘for m’’} // end of ‘‘for n’’ F igure Appendix B: More ellipse equations L , : ( x + y ) + = xy + x + y L , : (2 x + y ) + = xy + x + y L , : (3 x + y ) + = xy + x + y L , : (cid:32) x + y (cid:33) + = xy + x + y L , : (4 x + y ) + = xy + x + y L , : (cid:32) x + y (cid:33) + = xy + x + y L , : (cid:32) x + y (cid:33) + = xy + x + y L , : (cid:32) x + y (cid:33) + = xy + x + y L , : (cid:32) x + y (cid:33) + = xy + x + y L , : (5 x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + y ) + = xy + x + y L , : ( x + ) + = xy + x + y erzy Kocik Apollonian depth and the accidental fractal Appendix C: A remark on the “Apollonian”
Figure 9.5 shows a generic Apollonian disk packing (left) and a non-Apolloniandisk packing (only one of the two completions of the triple (a,b,c) belongs to thepacking.) a bc F igure An Apollonian disk packing is a collection of disks (some of possibly of non-positive curvature) such that the following
Apollonian rules are satisfied: (1) Itcontains a tricycle; (2) No disks overlap; (3) For any three mutually tangent circles,both Descartes solutions also belong to the packing.
Acknowledgments
I am indebted to Philip Feinsilver for his comments on this manuscript.
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] René Descartes, Oeuvres de Descartes, Correspondence IV, (C. Adam and P.Tannery,Eds.), Paris: Leopold Cerf 1901.[3] Lester R. Ford, Fractions, American Mathematical Monthly, (9) 45, 1938, pp.586â ˘A ¸S601.[4] Jerzy Kocik: A note on unbounded Apollonian disk packings,arXiv:1910.05924.[5] 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 / http://processingjs.org/ erzy Kocik Apollonian depth and the accidental fractal Additional images
Some color images that are too ink-draining to be part of the main body of thetext. Putting them here allows one to prevent printing them. F igure F igure F igure F igureigure