Soddy's hexlet 1

Soddy's hexlet

Figure 1. A family of hexlets related by a rotation and scaling. The centers of thespheres fall on an ellipse, making it an elliptic hexlet.

In geometry, Soddy's hexlet is a chain ofsix spheres (shown in grey in Figure 1),each of which is tangent to both of itsneighbors and also to three mutually tangentgiven spheres. In Figure 1, these threespheres are shown as an outercircumscribing sphere C (blue), and twospheres A and B (green) above and belowthe plane of their centers. In addition, thehexlet spheres are tangent to a fourth sphereD (red in Figure 1), which is not tangent tothe three others.

According to a theorem published byFrederick Soddy in 1937,[1] it is alwayspossible to find a hexlet for any choice ofmutually tangent spheres A, B and C.Indeed, there is an infinite family of hexletsrelated by rotation and scaling of the hexletspheres (Figure 1); in this, Soddy's hexlet isthe spherical analog of a Steiner chain of sixcircles.[2] Consistent with Steiner chains, thecenters of the hexlet spheres lie in a single plane, on an ellipse. Soddy's hexlet was also discovered independently inJapan, as shown by Sangaku tablets from 1822 in the Kanagawa prefecture.[3]

DefinitionSoddy's hexlet is a chain of six spheres, labeled S1–S6, each of which is tangent to three given spheres, A, B and C,that are themselves mutually tangent at three distinct points. (For consistency throughout the article, the hexletspheres will always be depicted in grey, spheres A and B in green, and sphere C in blue.) The hexlet spheres are alsotangent to a fourth fixed sphere D (always shown in red) that is not tangent to the three others, A, B and C.Each sphere of Soddy's hexlet is also tangent to its neighbors in the chain; for example, sphere S4 is tangent to S3 andS5. The chain is closed, meaning that every sphere in the chain has two tangent neighbors; in particular, the initialand final spheres, S1 and S6, are tangent to one another.

Soddy's hexlet 2

Annular hexlet

Figure 2: An annular hexlet.

The annular Soddy's hexlet is a special case (Figure 2), in whichthe three mutually tangent spheres consist of a single sphere ofradius r (blue) sandwiched between two parallel planes (green)separated by a perpendicular distance 2r. In this case, Soddy'shexlet consists of six spheres of radius r packed like ball bearingsaround the central sphere and likewise sandwiched. The hexletspheres are also tangent to a fourth sphere (red), which is nottangent to the other three.

The chain of six spheres can be rotated about the central spherewithout affecting their tangencies, showing that there is an infinitefamily of solutions for this case. As they are rotated, the spheres ofthe hexlet trace out a torus (a doughnut-shaped surface); in other words, a torus is the envelope of this family ofhexlets.

Solution by inversionThe general problem of finding a hexlet for three given mutually tangent spheres A, B and C can be reduced to theannular case using inversion. This geometrical operation always transforms spheres into spheres or into planes,which may be regarded as spheres of infinite radius. A sphere is transformed into a plane if and only if the spherepasses through the center of inversion. An advantage of inversion is that it preserves tangency; if two spheres aretangent before the transformation, they remain so after. Thus, if the inversion transformation is chosen judiciously,the problem can be reduced to a simpler case, such as the annular Soddy's hexlet. Inversion is reversible; repeatingan inversion in the same point returns the transformed objects to their original size and position.Inversion in the point of tangency between spheres A and B transforms them into parallel planes, which may bedenoted as a and b. Since sphere C is tangent to both A and B and does not pass through the center of inversion, C istransformed into another sphere c that is tangent to both planes; hence, c is sandwiched between the two planes a andb. This is the annular Soddy's hexlet (Figure 2). Six spheres s1–s6 may be packed around c and likewise sandwichedbetween the bounding planes a and b. Re-inversion restores the three original spheres, and transforms s1–s6 into ahexlet for the original problem. In general, these hexlet spheres S1–S6 have different radii.An infinite variety of hexlets may be generated by rotating the six balls s1–s6 in their plane by an arbitrary anglebefore re-inverting them. The envelope produced by such rotations is the torus that surrounds the sphere c and issandwiched between the two planes a and b; thus, the torus has an inner radius r and outer radius 3r. After there-inversion, this torus becomes a Dupin cyclide (Figure 3).

Soddy's hexlet 3

Figure 3: A Dupin cyclide, through which thehexlet spheres rotate, always touching. The

cyclide is tangent to an inner sphere, an outersphere and two spheres above and below the

"hole" in the "doughnut".

Dupin cyclide

The envelope of Soddy's hexlets is a Dupin cyclide, an inversion of thetorus. Thus Soddy's construction shows that a cyclide of Dupin is theenvelope of a 1-parameter family of spheres in two different ways, andeach sphere in either family is tangent to two spheres in same familyand three spheres in the other family.[4] This result was probablyknown to Charles Dupin, who discovered the cyclides that bear hisname in his 1803 dissertation under Gaspard Monge.[5]

Relation to Steiner chains

Figure 4: Steiner chain of six circlescorresponding to a Soddy's hexlet.

The intersection of the hexlet with the plane of its spherical centers producesa Steiner chain of six circles.

Parabolic and hyperbolic hexlets

It is assumed that spheres A and B are the same size.In any elliptic hexlet, such as the one shown at the top of the article, there aretwo tangent planes to the hexlet. In order for an elliptic hexlet to exist, theradius of C must be less than one quarter that of A. If C's radius is one quarterof A's, each sphere will become a plane in the journey. The inverted imageshows a normal elliptic hexlet, though, and in the parabolic hexlet, the pointwhere a sphere turns into a plane is precisely when its inverted image passesthrough the centre of inversion. In such a hexlet there is only one tangent plane to the hexlet. The line of the centresof a parabolic hexlet is a parabola.

If C is even larger than that, a hyperbolic hexlet is formed, and now there are no tangent planes at all. Label thespheres S1 to S6. S1 thus cannot go very far until it becomes a plane (where its inverted image passes through thecentre of inversion) and then reverses its concavity (where its inverted image surrounds the centre of inversion).Now the line of the centres is a hyperbola.The limiting case is when A, B and C are all the same size. The hexlet now becomes straight. S1 is small as it passesthrough the hole between A, B and C, and grows till it becomes a plane tangent to them. The centre of inversion isnow also with a point of tangency with the image of S6, so it is also a plane tangent to A, B and C. As S1 proceeds,its concavity is reversed and now it surrounds all the other spheres, tangent to A, B, C, S2 and S6. S2 pushes upwardsand grows to become a tangent plane and S6 shrinks. S1 then obtains S6's former position as a tangent plane. It thenreverses concavity again and passes through the hole again, beginning another round trip. Now the line of centres is adegenerate hyperbola, where it has collapsed into two straight lines.[2]

Soddy's hexlet 4

Sangaku tablets

Replica of Sangaku at Hōtoku museum inSamukawa Shrine.

The Japanese mathematicians analysed the packing problems in which circlesand polygons, balls and polyhedrons come into contact and often found therelevant theorems independently before their discovery by Westernmathematicians. The Sangaku about hexlet was made by Irisawa ShintarōHiroatsu in the family of Uchida Itsumi and dedicated to Samukawa Shrineon May, 1822. The original sangaku has been lost and recorded in theUchida's book of Kokinsankagami on 1832. The replica of the sangaku wasmade from the record and dedicated to Hōtoku museum in Samukawa Shrineon August, 2009.[6]

The sangaku by Irisawa consists of 3 problems and the third problem relatesto Soddy's hexlet: "the diameter of the outer circumscribing sphere is 30 sun.The diameters of the nucleus balls are 10 sun and 6 sun each. The diameter ofone of the balls in the chain of balls is 5 sun. Then I asked for the diametersof the remaining balls. The answer is 15 sun, 10 sun, 3.75 sun, 2.5 sun and2+8/11 sun."[7]

By his answer, the method to calculate the diameters of the balls is written down and can consider it the followingformulas to be given in the modern scale. If the ratio of the diameter of the outside ball to the nucleus balls are a1, a2,and if the ratio of the diameter to the chain balls are c1, ..., c6. I want to represent c2, ..., c6 by a1, a2, c1. If

then,

.

Then c1 + c4 = c2 + c5 = c3 + c6. If r1, ..., r6 are the diameters of six balls, then we get the formula:

Notes[1][1] Soddy 1937[2][2] Ogilvy 1990[3][3] Rothman 1998[4][4] Coxeter 1952[5] O'Connor & Robertson 2000[6] Dictionary of Wasan (Wasan no Jiten in Japanese), p.443[7] Sangaku Collection in Kanagawa prefecture (Kanagawa-ken Sangaku-syû in Japanese), pp.21-24.

Soddy's hexlet 5

References• Amano, Hiroshi (1992), Sangaku Collection in Kanagawa prefecture (Kanagawa-ken Sangaku-syū in Japanese),

Amano, Hiroshi.• Coxeter, HSM (1952), "Interlocked rings of spheres", Scripta Mathematica 18: 113–121.• Fukagawa, Hidetoshi; Rothman, Tony (2008), Sacred Mathematics: Japanese Temple Geometry (http:/ / press.

princeton. edu/ titles/ 8646. html), Princeton University Press, ISBN 978-0-691-12745-3• O'Connor, John J.; Robertson, Edmund F. (2000), "Pierre Charles François Dupin" (http:/ / www-groups. dcs.

st-and. ac. uk/ ~history/ Biographies/ Dupin. html), MacTutor History of Mathematics archive.• Ogilvy, C.S. (1990), Excursions in Geometry, Dover, ISBN 0-486-26530-7.• Soddy, Frederick (1937), "The bowl of integers and the hexlet", Nature (London) 139 (3506): 77–79,

doi:10.1038/139077a0.• Rothman, T (1998), "Japanese Temple Geometry", Scientific American 278: 85–91.• Yamaji, Katsunori; Nishida, Tomomi, ed. (2009), Dictionary of Wasan (Wasan no Jiten in Japanese), Asakura,

ISBN 978-4-254-11122-4.

External links• Weisstein, Eric W., " Hexlet (http:/ / mathworld. wolfram. com/ Hexlet. html)" from MathWorld.• B. Allanson. "Animation of Soddy's hexlet" (http:/ / members. ozemail. com. au/ ~llan/ soddy. html).• Japanese Temple Geometry (http:/ / www. ballstructure. com/ Japanese_Math/ J_Temple_Geometry. HTM) - The

animation 0 of SANGAKU PROBLEM 0 shows the case which the radiuses of spheres A and B are equal eachother and the centers of spheres A, B and C are on the line. The animation 1 shows the case which the radiuses ofspheres A and B are equal each other and the centers of spheres A, B and C are not on the line. The animation 2shows the case which the radiuses of spheres A and B are not equal each other. The animation 3 shows the casewhich the centers of spheres A, B and C are on the line and the radiuses of spheres A and B are variable.

• Replica of Sangaku at Hōtoku museum in Samukawa Shrine (http:/ / www. wasan. earth. linkclub. com/kanagawa/ samukawa. html) - The third problem relates to Soddy's hexlet.

Article Sources and Contributors 6

Article Sources and ContributorsSoddy's hexlet  Source: http://en.wikipedia.org/w/index.php?oldid=510382988  Contributors: 4, Adam majewski, AugPi, David Eppstein, Double sharp, Eewild, Geometry guy, Giftlite,GregorB, Lanthanum-138, Michael Hardy, N Shar, Rautermann, Rgdboer, Rjwilmsi, Terrek, Ttwo, TutterMouse, WillowW, 3 anonymous edits

Image Sources, Licenses and ContributorsImage:Rotating hexlet equator opt.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Rotating_hexlet_equator_opt.gif  License: Creative Commons Attribution-Sharealike 3.0 Contributors: WillowWImage:Annular Soddy hexlet.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Annular_Soddy_hexlet.jpg  License: Creative Commons Attribution-Sharealike 3.0  Contributors:WillowWImage:Cyclide.png  Source: http://en.wikipedia.org/w/index.php?title=File:Cyclide.png  License: GNU Free Documentation License  Contributors: Anarkman, SvdmolenImage:Steiner chain animation opt.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Steiner_chain_animation_opt.gif  License: Creative Commons Attribution-Sharealike 3.0 Contributors: WillowWImage:Sangaku of Soddy's hexlet in Samukawa Shrine.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Sangaku_of_Soddy's_hexlet_in_Samukawa_Shrine.jpg  License: CreativeCommons Attribution-Sharealike 3.0,2.5,2.0,1.0  Contributors: Shikishima Ken-ichi (talk)

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