AApollonian coronas and a new Zeta function
Jerzy Kocik
Department of Mathematics, Southern Illinois University, Carbondale, [email protected] 24, 2019
Abstract
We find a formula for the area of disks tangent to a given disk in an Apolloniandisk packing (corona) in terms of a certain novel arithmetic Zeta function. Theidea is based on “tangency spinors” defined for pairs of tangent disks.
Keywords:
Apollonian disk packing, spinor, Pythagorean triples, Euclid’sparametrization, Epstein Zeta function, Minkowski space, corona.
MSC:
1. Introduction
Apollonian disk packing is an arrangement of infinite number of disks determinedby three mutually tangent disks through recurrent inscribing new disks in triangle-like spaces between disks as they appear in this process. An Apollonian disk pack-ing is called integral if the curvatures of all disks are integers. If any four mutuallytangent disks (called disks in
Descartes configuration ) are integral, so is the wholepacking. See Figure 1 for two basic examples, the Apollonian Window [11] and theApollonian Belt.
66 66
12 1212 12 F igure
1: Apollonian Window (left) and Apollonian Belt (right)
Note that in both arrangements, the disks – if tangent – are tangent externally. Thegreatest circle in the Apollonian Window is understood as the boundary of the diskthat lies outside of it; consequently its radius and curvature is equal to − a r X i v : . [ m a t h . M G ] S e p erzy Kocik Coronas, spinors, and a Zeta function
01 23 11 43 21 F igure
2: Ford circles than 1. Similarly, the lines in the Apollonian Belt are boundaries if half-planesthat lie outside of the central belt and are viewed as disks of curvature 0. The in-tegral packings are interesting due to their evident connection with number theory[8, 7, 12, 14, 15].Let us define a corona of a particular disk in an Apollonian disk packing A asthe set of all disk that are tangent to A . Our point of interest is the area of the coronasor its fragments. The famous Ford circles [3] can be interpreted as a corona in theApollonian Belt composed of circles tangent to 0-curvature disk at the bottom (seeFigure 2). The area of the disks between two large circles was found in [2]. Quiteremarkably, it is related to Riemann zeta functions: A Ford = π (cid:32) + ζ (3) ζ (4) (cid:33) (1.1)In this paper we present a formula for coronas and their fragments in an arbitraryApollonian disk packing. To give a taste of its arithmetic elegance, here is a specificresult: the corona made of the disks tangent to the greatest circle (of radius 1, seeFigure 1, left) of the Apollonian Window consists of four congruent quarters. Thearea of each is: A AW = π (cid:88) ≤ k ≤ n gcd( n , k ) = (cid:0) n + k + (cid:1) (1.2)where the sum is restricted to coprimes ( n , k ). The greatest common divisor will bedenoted ( m , n ). One easily recognizes the largest disks among the first summands: A AW = (cid:0) + + (cid:1) + (cid:0) + + (cid:1) + (cid:0) + + (cid:1) + ... = + + + ... Section 3 provides the general formula for coronas in arbitrary Apollonian diskpackings. The formula defines a new type of arithmetic function quite similar toEpstein Zeta function, except for a free term (1 in the above formula) and, moreimportantly, the sum being taken over coprimes instead of all pairs of integers.The idea is based on the existence of the “spinor structure” in the Apolloniandisk packing. erzy Kocik Coronas, spinors, and a Zeta function
2. Spinors in Apollonian disk arrangements
The section summaries the essential facts extracted from [11].Every disk in the Cartesian plane may be given a symbol , a fraction-like labelthat encodes the size and position of the disk: the curvature (reciprocal of radii) isindicated in the denominator while the positions of the centers may be read o ff byinterpreting the symbol as a pair of fractions [10]. For example: symbol: , = ⇒ radius: r = center: ( x , y ) = (cid:16) , (cid:17) = (cid:16) , (cid:17) The numerator will be called the reduced coordinates of the a disk’s center anddenoted by dotted letters ( ˙ x , ˙ y ) = ( x / r , y / r ). Unbounded disk extending outside acircle are given negative radius and curvature. Quite remarkably, all symbols in theApollonian Window have integer entries (see Figure 3). , –1 (Outer disk)
3, 4 -8,12 23
15 8
0, 2 3 -21,20 30 -1,0 2 -3, 4 -8,6 11 -5,12 -15 8 -24,10 27 Figure 3.1 : Four circles in a Descartes configuration — special cases.
Figure 3.2:
Apollonian window (upper half) with circles labeled by “fractions”
C A B C D A B C D A D B C (a ) (b ) (c ) (d ) A B D
Figure 2.4:
In the 5th century BCE Pappus noticed that each of the circles in the chain inscribed into an arbelos, the shaded ideal triangle formed by three mutually tangent circles with the centers on a line, is an integer number of its diameter above that line. Then n -th circle is n times above. F igure
3: Symbols in the Apollonian Window (upper half).
The symbols in an Apollonian disk packing can be generated from the first threevia the Descartes theorem. It states that the curvatures of four mutually tangentdisks, the so-called
Descartes configuration , satisfy the Descartes formula (1643)[5, 18]: ( A + B + C + D ) = A + B + C + D ) (2.1)A more convenient version of this quadratic formula is D + D (cid:48) = A + B + C ) (2.2)where D and D (cid:48) are the two solutions to (2.1), given A , B , C . The last formulaholds for the corresponding reduced positions of the centers of the disks (consult[9]). Since an Apollonian gasket is a completion of a Descartes configuration, allsymbols may be derived from the first four using (2.2) . Thus integrality of thesymbols in the Apollonian Window follows from the integrality of the first fourdisks. erzy Kocik Coronas, spinors, and a Zeta function F igure
4: Pythagorean triples in the Apollonian Window
Every pair of tangent disks in A defines a right triangle with sides proportionalto a Pythagorean triple as follows:˙ x , ˙ y β (cid:90) ˙ x , ˙ y β (cid:55)→ abc ≡ β ˙ x − β ˙ x β ˙ y − β ˙ y β + β (2.3)where a + b = c (see Figure 4 for examples). Note that in the case of the Apol-lonian Window, all of these Pythagorean triples are integral. The actual size of thetriangle in the plane is scaled down by the factor of β β (gray triangles in Figure4). The next step is to recall that Pythagorean triples admit Euclidean parametersthat determine them via the following prescription: u = [ m , n ] → ( a , b , c ) = ( m − n , mn , m + n ) (2.4)(see, e.g., [17, 19]). This map has a simple form in terms of complex numbers: u = m + ni → u = a + bi = ( m − n ) + mn i (2.5)with c = | u | = m + n . As explained in [10], Euclidean parameters can be viewedas a spinor , a vector u ∈ R . Hence our concept of tagency spinor defined for anordered pair of two tangent disks (not necessarily integral) of curvatures A and B asas : u = ± (cid:114) zAB (2.6)where z ∈ C is the complex number reprezenting vector joining the centers of thedisks. We view spinor as a vector in a two dimensional Euclidean space and use thecomplex structure as a convenient description, u ∈ C (cid:27) R via identification (cid:20) mn (cid:21) ≡ m + ni erzy Kocik Coronas, spinors, and a Zeta function Remark:
A Pythagorean triangle with sides satisfying a + b − c = R , and, as such, it may berepresented as the tensor square of a spinor from the associated two-dimensionalspinor space R .
11 Here however the quadratic form for Pythagorean triples is the determinant c – a – b , and the spinor space is equipped with a Euclidean structure in order to maintain the Kronecker product on the right side geometrically meaningful.
6. Spinors in the Apollonian window
Recapitulating the process: → Pick two tangent circles in A Figure 6.1:
A spinor in the Apollonian window. Although the true dimensions of the shaded triangle are (5/22, 12/22, 12/22), one takes the triple (5,12,13) to define the spinor at the tangency to be, up to sign: [3,2] or [-3,-2].
For the other order of circles, from small to large, the triangle reads (-5/22, -12/22, 13/22) and the spinor is [2,-3] or [-2, 3] ±±±± [3,2] ±±±± [-2,3]
5, 12, 13 22 F igure
5: Spinors in the Apollonian Window
In graphical representation we shall mark a spinor by an arrow that indicates theorder of disks, and will label it by its value. Note that the spinor is defined up to asign, since ( − u ) = u . Also, the spinor depends on the order of the circles: if u is aspinor for ( AB ), then the spinor for ( BA ) is u (cid:48) = iu (up to a sign).Figure 6 shows spinors in the Apollonian Window A (for visual clarity thebrackets are omitted). There are four vectors at every point of tangency: two signstimes two orderings of circles. All spinors in the Apollonian Window A are integralbut we will not restrict our considerations to the integral examples.
6 2 6 2 -4 1 -5 2 0 -1 0 -1 1 0 -2 0 1 0 -2 -1 24 -1 -1 -2 2 -1 -2 3 0 F igure
6: Flow of spinors in the Apollonian Window
The key property of spinors is that in a quartet of mutually tangent disks (disks erzy Kocik Coronas, spinors, and a Zeta function ± ) such that these twoproperties hold “curl u =
0” : u + u + u = u =
0” : u + u + u = u i j represents a spinor for i -th and j -th tangent disks. These results may beviewed as am “underground” version of Descartes’ theorem on circles.These laws have local character. Extending the appropriate choice of signs toa greater system, like the whole Apollonian gasket, will encounter topological ob-structions.In the remainder of this section we review the main properties of tangency spinors;for proofs see [11]. The capital letters will denote both circles and their curvatures. Proposition 2.1.
If u is the tangency spinor for two tangent disks of curvatures Aand B, respectively, (Figure 7, left) then | u | = A + B (2.7) B B c C A C A u B A a b a b F igure
7: Left: Two curvatures and a spinor; Center: curvature from two spinors; Right:curl u = Theorem 2.2 ( curvatures from spinors). In the system of three mutually tangentcircles, the symplectic product of two spinors directed outward from (respectivelyinward into) one of the circles equals (up to sign) its curvature, e.g., following no-tation of Figure 7, center: B = ± a × b (2.8) where a × b : = det[ a | b ] = a b − a b . Theorem 2.3 ( spinor curl). The signs of the three tangency spinors between bethree mutually tangent circles (Figure 7, right) may be chosen so thata + b + c = u = ” ] (2.9) erzy Kocik Coronas, spinors, and a Zeta function Theorem 2.4.
Let A, B, C, and D be four circles in a Descartes configuration. [Vanishing divergence]:
If a, b and c are tangency spinors for pairs AD, BD andCD (Figure 8 left), then their signs may be chosen so thata + b + c = “ div u = ” ] (2.10) The same property holds for the outward oriented spinors. [Additivity]:
If a and b are tangency spinors for pairs CA and CB ( Figure 8 right),then there is a choice of signs such that the sumc = a + b (2.11) is a spinor of tangency for CD. a b c a b a+b A B C F igure
8: (a) vanishing divergence, (b) spinor addition
Theorem 2.4 can be iterated to produce spinors for all circles inscribed betweentwo initial circles.The last m concerns the signs of the spinors. Let A , B and C be three mutuallytangent disks, and let D and D (cid:48) be two disks that complete this triple to respectiveDescartes configurations. Consider spinors a , b and d from circle C to circles A , B ,and D , like in the Figure 8, with some fixed singes. D is the unlabeled small diskin the center. Spinors a to b are harmonized over the arc of circle C through D if a + b = ± d for some choice of the sign of d . If a and b are harmonized, so are( − a ) and ( − b ). If (signs of ) spinors are not harmonized, then spinors a and ( − b areharmonized over the complementary part of the circle C ,.i.e., the arc tangent to D (cid:48) .Here is an example referring to Figure 6: Spinors (cid:20) (cid:21) and (cid:20) (cid:21) are harmonizedover the upper arc of disk “2” since (cid:20) (cid:21) + (cid:20) (cid:21) = (cid:126)
34. But (cid:20) (cid:21) and (cid:20) − − (cid:21) are not, they areharmonized over the lower arc since (cid:20) (cid:21) + (cid:20) − − (cid:21) = − (cid:20) (cid:21) , a spinor towards the externaldisk “ −
1, which is the other solution to Descartes problem for disks (2 , , erzy Kocik Coronas, spinors, and a Zeta function
3. Corona’s area
Here is the main result:
Theorem 3.1.
Let u and v be two spinors oriented from a disk of curvature B inan Apollonian disk packing towards two mutually tangent disk in its corona. Definematrix M, and a “dummy” integer vector f as: u = (cid:34) ab (cid:35) , v = (cid:34) cd (cid:35) , M = (cid:34) a cb d (cid:35) , f = (cid:34) mn (cid:35) Then the area of the entire corona isA ( M ) = π (cid:88) f ∈ Z (cid:0) (cid:107) M f (cid:107) − B (cid:1) − = π (cid:88) m , n ∈ Z ( m , n ) = (cid:0) ( am + cn ) + ( bm + dn ) − B (cid:1) (3.1) Towards the proof.
Suppose we want to find the area of a fragment of coronaconsisting of disks between the circles A and B around the base circle of curvature C in Figure 8. We can recover all spinors between a and b using recursively Theorem2.4B. in a form of a series of sequences R i , each obtained from the previous byinserting new terms that are sums of the neighboring terms of the previous. A fewinitial rows are shown below: R : a b R : a a + b b R : a a + b a + b a + b b R : a a + b a + b a + b a + b a + b a + b a + b b (If R ni denotes the i-th term in the n-th row, the iteration is defined by R n + , i = R n , i and R n + , i + = R n , i + R n , i + , where i = , ..., n ). Such a system of sequences willbe called a (generalized) Stern-Brocot array. After removal of the repetitious termsalong the columns, it becomes a Stern-Brocot tree.Each of the spinors u from this collection determines the curvature of the asso-ciated circle tangent to C : curv = (cid:107) u (cid:107) − C (3.2)by Proposition 2.1, Eq. (2.7). The corresponding area is area = π ( curv ) and the total area is the sum over all entries produced by the above scheme (metaphor-ically, all entries in the “last”, limit, row R ∞ ). erzy Kocik Coronas, spinors, and a Zeta function
22 33
66 66 (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) F igure
9: Apollonian Window – right op corner fragment the major corona
Let us consider a particular example of the corona around the circle of curvature2 in the Apollonian Window, shown in Figure 9. If we start with vectors a = [1 2] T and b = [2 2] T , the Stern-Brocot array looks like the one below, left: (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) -2-2 (cid:21)(cid:20) (cid:21) (cid:20) -10 (cid:21) (cid:20) -2-2 (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) -10 (cid:21) (cid:20) -3-2 (cid:21) (cid:20) -2-2 (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) -12 (cid:21) (cid:20) -10 (cid:21) (cid:20) -4-2 (cid:21) (cid:20) -3-2 (cid:21) (cid:20) -5-4 (cid:21) (cid:20) -2-2 (cid:21) F igure
10: Examples of Stern-Brocot arrays
Spinor u = [ m n ] T determines curvature curv = m + n − , , , , , , , , erzy Kocik Coronas, spinors, and a Zeta function
22 33
66 66 (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) F igure
11: Apollonian Window – right op corner fragment the major corona of the corona is
Area = (cid:88) π ( m + n − where the sum is over all ( m , n ) that appear in the Stern-Brocot produced by theabove tree. The sum of terms over a particular row R n gives the area of the circlesup to the n -th depth.To recover the full corona around the disk one may want to repeat the sameprocess with one of the initial spinors changed to its negative, to account for thecomplementary segment of the corona (see the box on the right in Figure 10).The sum would have to be over a set of vectors reproduced case-by-case. This issu ffi cient for estimation but it lacks an elegant algebraic form. But there is a remedy,as presented in the following. B. Universal case.
The upper half of the corona in Apollonian Window around thegreat circle of curvature ( − (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) F igure
12: Universal Stern-Brocot arrayerzy Kocik Coronas, spinors, and a Zeta function F igure
13: Euclidean orchard and its extensions.
This arrangement proves to be universal for our purpose. It has two essentialproperties: (1) the coe ffi cients k and n of every entry [ k n ] T are co-primes, and (2)every coprime pair appears in the tree. This follows from the fact that if we interpretthe vector as a fraction, (cid:20) kn (cid:21) (cid:55)→ k / n , the above tree becomes the well-known classicalStern-Brocot tree of positive rational numbers, which is known to lists them in theform of reduced fractions, each exactly once (one must ignore the repetitions downalong the columns).The vectors of Stern-Brocot tree of spinors may be visualized as so-called Eu-clid’d orchard (see Fig 13, left). To account also for the bottom part of the Apollo-nian Window, we must extend the set to include spinors produced by the pair (cid:34) (cid:35) , (cid:34) − (cid:35) . This would graphically correspond to the “extended orchard” shown in Figure 13,center. Note however that one entry is repeated, since [ − , T ∼ [1 , T and bothcorrespond to the same disk. Instead of removing one copy, extend the orchard toall pairs of co-primes in the lattice Z × Z , as shown in Figure 13, right. Now everyentry appears exactly twice. We have take half of the sum, but we gain more elegantformulation: Proposition 3.2.
The total area of disks tangent to the great circle in the Apollonianwindow is A = π (cid:88) m , n ∈ Z gcd( m , n ) = (cid:0) m + n + (cid:1) where m and n run over all pairs coprime integers (including negative). Denote Z the subset of the integral lattice Z × Z restricted to vectors with co-prime coe ffi cients: Z = (cid:26) (cid:20) mn (cid:21) ∈ Z × Z (cid:12)(cid:12)(cid:12) gcd( m , n ) = (cid:27) (3.3)(it can be called the “extended Euclid’s orchard”). The above formula may now berewritten as A = (cid:88) v ∈ Z (cid:0) (cid:107) v (cid:107) + (cid:1) erzy Kocik Coronas, spinors, and a Zeta function C. General case.
Now, to get a formula for an arbitrary corona, notice that theStern-Brocot tree determined by vectors u = (cid:34) ab (cid:35) , v = (cid:34) cd (cid:35) , may be obtained from the universal Stern-Brocot tree (Figure 12) by replacing itsevery entry by a vector transformed by matrix M = [ u | v ] ≡ (cid:34) a cb d (cid:35) ∈ End( n , R ) (cid:27) Mat × ( R ) (3.4)For instance, the Stern-Brocot array in Figure 10 may be viewed as the product ofthe map Z o (cid:51) (cid:34) mn (cid:35) (cid:55)→ (cid:34) (cid:35) (cid:34) mn (cid:35) ∈ R Hence our main result:
The Main Theorem [Same as Thm. 3.1]:
The area of the corona of the disk B with two adjacent spinors forming matrix M (see (3.4)) isEntire corona: K ( M ) = (cid:88) f ∈ Z (cid:0) (cid:104) f T | M T M | f (cid:105) − B (cid:1) − (3.5)To account for a fragment of corona between the circles related to spinors u and v , we use the same formula except with the summation going over one of these sets( a ) N o = : { ( m , n ) ∈ N × N | gcd( m , n ) = } ( b ) ˙ N o = : { ( m , n ) ∈ N × N | gcd( m , n ) = } where N = { , , ... } and ˙ N o = { , , , ... } . Case (a) captures disks strictly betweenthe end-disks, while (b) includes the end disks (see Figure 13, left).The formula for the total corona area A involves a choice two spinors, yet itsvalue is is invariant with respect to this choice. Indeed, the group SL(2 , Z ) acts on Z o as bijection and permutes spinors, preserving their mutual relations of neighbor-hood. Thus the matrix M is transformed accordingly: M = [ u | v ] (cid:55)→ [ u | v ] g = Mg Proposition 3.3. (Invariance). The area function of the corona is invariant underaction of the modular group, that is:K ( M ) = K ( gM ) for any g ∈ SL(2 , Z ) . erzy Kocik Coronas, spinors, and a Zeta function Proof:
Denote the summand in (3.1) by A ( M , f , B ). Then (cid:88) f ∈ Z A (cid:2) ( Mg ) f (cid:3) = (cid:88) f ∈ Z A (cid:2) M ( g f ) (cid:3) = (cid:88) g f ∈ Z A [ M f ] = (cid:88) f ∈ g − ( Z ) A [ M f ] = (cid:88) f ∈ Z A [ M f ] (cid:3) The main theorem is equivalent to the following statement:
Proposition 3.4.
Let u and v be two neighboring spinors in a corona around a diskof curvature B in an Apollonian disk packing. Maintain the previous notation: u = (cid:34) ab (cid:35) , v = (cid:34) cd (cid:35) , M = [ u | v ] ≡ (cid:34) a cb d (cid:35) , f = (cid:34) mn (cid:35) Then the quadratic polynomialP ( m , n ) = f T ( M T M ) f − Breproduces all curvatures in the corona of B with the integers m , n ∈ Z running ofall pairs of integer co-primes, including ( ± , and (0 , ± . Each curvature appearsin the range of the polynomial twice due to P ( f ) = P ( − f ) . In terms of the entries of the matrices, the polynomial is P ( m , n ) = ( a + c ) m + ( c + d ) n + ( ab + cd ) mn − B The invariance of the range of the above polynomial with respect to the groupSL(2 , Z holds for the same argument as in Proposition 3.3. erzy Kocik Coronas, spinors, and a Zeta function
4. Examples and special cases
Example 1: Ford circles . Set the Apollonian Belt vertically, as in Figure 14. Thespinors at the two unit circles are v = (cid:34) (cid:35) and w = (cid:34) (cid:35) The general formula (3.1) becomes A = π (cid:88) gcd( m , k ) = m + k ) = π ∞ (cid:88) n = ϕ ( n ) n where ϕ ( n ) denotes the Euler’s totient function (the number of positive coprimeswith n not exceeding n ). The last equation is implied by a simple – easy to show –property true for n > ϕ ( n ) = | { ( i , j ) ∈ N × N (cid:12)(cid:12)(cid:12) i + j = n , gcd( i , j ) = } | The result is an example of Dirichlet series, which is known to converge to a ratioof Riemann zeta functions [4]. In general: ∞ (cid:88) n = ϕ ( n ) n s = ζ ( s − ζ ( s ) } } }
99 99 99 (cid:104) (cid:105)(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)(cid:104) (cid:105) (cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105)(cid:104) (cid:105) F igure
14: Ford circles as a corona in the Apollonian Belterzy Kocik Coronas, spinors, and a Zeta function
22 33
66 66 (cid:34) (cid:35)(cid:34) (cid:35) F igure
15: Apollonian Window – right op corner fragment the major corona
Hence, in our case: A Ford /π = + ∞ (cid:88) n = ϕ ( n ) n = + ζ (3) ζ (4) Example 2: Apollonian Window – a quarter.
Consider the upper right quarter ofthe corona of the great circle in the Apollonian Window, Fig. 15. The spinors at thetwo big circle are v = (cid:34) (cid:35) , w = (cid:34) (cid:35) , B = − A v , w = ∞ (cid:88) n = (cid:88) k ≤ n ( n , k ) = π (cid:0) n + k + (cid:1) Example 3: Apollonian Window – around b = . Consider the corona around theright disc of curvature B = v = (cid:34) (cid:35) and w = (cid:34) (cid:35) ⇒ A v , w = ∞ (cid:88) n = (cid:88) k ≤ n ( n , k ) = (cid:0) n + k − (cid:1) Similarly, the fragment between the circles of curvature 3 and ( −
1) (the one thatincludes the chain 6, 11, 18 etc) can be expressed by spinors v = (cid:34) (cid:35) w = (cid:34) (cid:35) ⇒ A v , w = (cid:88) k , n ∈ N ( n , k ) = (cid:0) n + nk + k − (cid:1) erzy Kocik Coronas, spinors, and a Zeta function T and [2 0] T , we get A = (cid:88) k , n ∈ N gcd( n , k ) = (cid:0) n + k − (cid:1) Example 4:
Here is example an involving a non-symmetric disk packing generatedby a Descartes configuration of disk of curvatures ( − , , , − , − − , , , , , , The resulting spinors are also integral. Figure 16 illustrates the packing and anumber of spinors that originate at the disk of curvature B =
24. Pick a pair ofneighboring spinors v = (cid:34) (cid:35) and w = (cid:34) − (cid:35) The corona around that circle can evaluated as A = π (cid:88) k , n ∈ N gcd( n , k ) = (cid:0) n + kn + k − (cid:1)
132 132 (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) − (cid:35) (cid:34) (cid:35) (cid:34) (cid:35) F igure
16: Apollonian disk packing ( − , , ,
28) and some spinorserzy Kocik Coronas, spinors, and a Zeta function
5. Comparison with Epstein zeta function
Recall the definition of the Epstein zeta function [1, 4]: Z ( S , ρ ) = (cid:88) v ∈ Z n \{ } (cid:16) v T S v (cid:17) − ρ , (5.1)where S is the n × n matrix S of a positive definite real quadratic form, and ρ is acomplex variable (the real part of which with is greater than n / Definition 5.1.
A geometric zeta function is Z ◦ ( S , ρ, β ) = (cid:88) v ∈ Z n E (cid:16) v T S v − β (cid:17) − ρ , (5.2)where where S is any n × n matrix S of a positive definite real quadratic form and ρ a complex variable. The sum runs over all column vectors with co-prime integerentries, i.e., over the “ n = dimensional Euclidean orchard”: Z E = { v = [ x , ..., x n ] T ∈ Z n | gcd( x , ..., x n ) = } Additional condition of geometric compatibility that may be imposed is β = ± det S Here is justification for this definition. First note that, referring to (3.1), (cid:107) M u (cid:107) = ( M u ) T ( M u ) = u T ( M T M ) u where S = M T M is evidently symmetric positive definite matrix. Substituting S = M T M , n = B = ± det M , and ρ = Area = π Z ◦ ( M T M , , β ) ( β = ± det S )Note the di ff erence the two Zeta functions: (1) our sum runs over pairs of co-primes, and (2) a shift by β is present in the formula. Analytical properties andpossible other arithmetic significance of the new Zeta function remain to be studied. erzy Kocik Coronas, spinors, and a Zeta function Appendix A: Practical formulas
In order to evaluate the area of a corona in a recursive way one may organize thesum in a diagonal order of counting. Suppose a disk of curvature B has a pair ofneighboring spinors v = (cid:34) ab (cid:35) , w = (cid:34) cd (cid:35) , N = some big numberThen the area between the two disks (end-disk inclusive) is K ( v , w ) = N (cid:88) n = n (cid:88) k = coprime( n , k ) (cid:0) ( ka + ( n − k ) c ) + ( kb + ( n − k ) d ) − B (cid:1) = N (cid:88) n = n (cid:88) k = coprime( n , k ) (cid:18) (cid:13)(cid:13)(cid:13)(cid:13) (cid:104) a bc d (cid:105) (cid:104) −
10 1 (cid:105) (cid:104) km (cid:105) (cid:13)(cid:13)(cid:13)(cid:13) − B (cid:19) where the function coprime( n , k ) returns 1 if n and k are coprimes, and 0 otherwise.Total area of the whole corona may be evaluated by combining the sums over twoarcs of the disk: K ( v , w ) = a + b − B ) + c + d − B ) + (cid:80) Nn = (cid:80) n − k = coprime( n , k ) (cid:0) ( am + cn ) + ( bm + dn ) − B (cid:1) + coprime( n , k ) (cid:0) ( am − cn ) + ( bm − dn ) − B (cid:1) Clearly, the actual value is the limit N → ∞ of the above expressions. References [1] Epstein, P. Zur Theorie allgemeiner Zetafunktionen. I.,
Math. Ann. , 614-644, 1903.[2] G.T. Williams and D.H. Brown, A family of integers and a theorem on circles,Amer. Math. Monthly 54 (1947), no 9, 534–536.[3] L.R. Ford, Fractions, Amer. Math., Monthly 45 (1938) no 9, 586–601.[4] Hardy, G. H., and Wright, E. M. (2008), An Introduction to the Theory ofNumbers (Sixth ed.), Oxford University Press (Sec. 17.5)[5] Donald Coxeter, The Problem of Apollonius, The American MathematicalMonthly , 1968, , (1) 1968, pp. 5-15.[6] Lester R. Ford, Fractions, American Mathematical Monthly , (9) 45, 1938, pp586–601.[7] Ronald L. Graham, Je ff rey C. Lagarias, Colin L. Mallows, Allan R. Wilks andCatherine H. Yan, Apollonian circle packings: geometry and group theory I.Apollonian group, Discrete & Computational Geometry
34 (2005), 547–585[arXiv.org / math.MG / erzy Kocik Coronas, spinors, and a Zeta function J. Lond. Math. Soc. , 42 (1967),281–291.[9] Jerzy Kocik, A matrix theorem on circle configuration (arXiv:0706.0372v2).(available at http: // arxiv.org / PS_ cache / hep-ph / pdf / / ff ord Algebras and Euclid’s Parameterization ofPythagorean Triples, Advances in Appl. Cli ff . Alg. / kocik / papers / ff .pdf][11] Jerzy Kocik, Spinors and Descartes configuration of disks,arXiv:1909.06994[math.MG][12] 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 / ff ord Algebras and Spinors (London Mathematical SocietyLecture Note Series), Cambridge University Press, 2 ed., 2001.[14] Benoit Mandelbrot: The Fractal Geometry of Nature, Freeman (1982)[15] Zdzisław A. Melzak, Infinite packings of disks. Canad. J. Math.
18 (1966),838–853.[16] Ian R. Porteous, Topological Geometry, Cambridge University Press; 2nd ed.1981.[17] Wacław Sierpi´nski, Pythagorean triangles,
The Scripta Mathematica Studies ,No 9, Yeshiva Univ., New York, 1962.[18] Fredric Soddy, The Kiss Precise.
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