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| [[Image:Rotating hexlet equator opt.gif|thumb|right|300px|Figure 1. A family of hexlets related by a rotation and scaling. The centers of the spheres fall on an [[ellipse]], making it an elliptic hexlet.]]
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| In [[geometry]], '''Soddy's hexlet''' is a chain of six [[sphere]]s (shown in grey in Figure 1), each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, these three spheres are shown as an outer circumscribing sphere (blue), and two spheres (not shown) above and below the plane the centers of the hexlet spheres lie on. In addition, the hexlet spheres are tangent to a fourth sphere (red in Figure 1), which is not tangent to the three others.
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| According to a [[theorem]] published by [[Frederick Soddy]] in 1937,<ref>{{Harvnb|Soddy|1937}}</ref> it is always possible to find a hexlet for any choice of mutually tangent spheres ''A'', ''B'' and ''C''. Indeed, there is an infinite family of hexlets related by rotation and scaling of the hexlet spheres (Figure 1); in this, Soddy's hexlet is the spherical analog of a [[Steiner chain]] of six circles.<ref name="ogilvy">{{Harvnb|Ogilvy|1990}}</ref> Consistent with Steiner chains, the centers of the hexlet spheres lie in a single plane, on an ellipse. Soddy's hexlet was also discovered independently in Japan, as shown by [[Sangaku]] tablets from 1822 in the Kanagawa prefecture.<ref>{{Harvnb|Rothman|1998}}</ref>
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| ==Definition==
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| Soddy's hexlet is a chain of six spheres, labeled ''S''<sub>1</sub>–''S''<sub>6</sub>, 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 hexlet spheres will always be depicted in grey, spheres ''A'' and ''B'' in green, and sphere ''C'' in blue.) The hexlet spheres are also tangent to a fourth fixed sphere ''D'' (always shown in red) that is not tangent to the three others, ''A'', ''B'' and ''C''.
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| Each sphere of Soddy's hexlet is also tangent to its neighbors in the chain; for example, sphere ''S''<sub>4</sub> is tangent to ''S''<sub>3</sub> and ''S''<sub>5</sub>. The chain is closed, meaning that every sphere in the chain has two tangent neighbors; in particular, the initial and final spheres, ''S''<sub>1</sub> and ''S''<sub>6</sub>, are tangent to one another.
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| ==Annular hexlet==
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| [[Image:Annular Soddy hexlet.jpg|thumb|right|200px|Figure 2: An annular hexlet.]]
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| The annular Soddy's hexlet is a special case (Figure 2), in which the three mutually tangent spheres consist of a single sphere of radius ''r'' (blue) sandwiched between two parallel planes (green) separated by a perpendicular distance 2''r''. In this case, Soddy's hexlet consists of six spheres of radius ''r'' packed like ball bearings around the central sphere and likewise sandwiched. The hexlet spheres are also tangent to a fourth sphere (red), which is not tangent to the other three.
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| The chain of six spheres can be rotated about the central sphere without affecting their tangencies, showing that there is an infinite family of solutions for this case. As they are rotated, the spheres of the hexlet trace out a [[torus]] (a doughnut-shaped surface); in other words, a torus is the [[envelope (mathematics)|envelope]] of this family of hexlets.
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| ==Solution by inversion==
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| The general problem of finding a hexlet for three given mutually tangent spheres ''A'', ''B'' and ''C'' can be reduced to the annular case using [[Inversive geometry|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 sphere passes through the center of inversion. An advantage of inversion is that it preserves tangency; if two spheres are tangent 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; repeating an inversion in the same point returns the transformed objects to their original size and position.
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| Inversion in the point of tangency between spheres ''A'' and ''B'' transforms them into parallel planes, which may be denoted as ''a'' and ''b''. Since sphere ''C'' is tangent to both ''A'' and ''B'' and does not pass through the center of inversion, ''C'' is transformed into another sphere ''c'' that is tangent to both planes; hence, ''c'' is sandwiched between the two planes ''a'' and ''b''. This is the annular Soddy's hexlet (Figure 2). Six spheres ''s''<sub>1</sub>–''s''<sub>6</sub> may be packed around ''c'' and likewise sandwiched between the bounding planes ''a'' and ''b''. Re-inversion restores the three original spheres, and transforms ''s''<sub>1</sub>–''s''<sub>6</sub> into a hexlet for the original problem. In general, these hexlet spheres ''S''<sub>1</sub>–''S''<sub>6</sub> have different radii.
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| An infinite variety of hexlets may be generated by rotating the six balls ''s''<sub>1</sub>–''s''<sub>6</sub> in their plane by an arbitrary angle before re-inverting them. The envelope produced by such rotations is the [[torus]] that surrounds the sphere ''c'' and is sandwiched between the two planes ''a'' and ''b''; thus, the torus has an inner radius ''r'' and outer radius 3''r''. After the re-inversion, this torus becomes a [[Dupin cyclide]] (Figure 3).
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| [[Image:Cyclide.png|thumb|left|Figure 3: A Dupin cyclide, through which the hexlet spheres rotate, always touching. The cyclide is tangent to an inner sphere, an outer sphere and two spheres above and below the "hole" in the "doughnut".]]
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| ==Dupin cyclide==
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| The [[envelope (mathematics)|envelope]] of Soddy's hexlets is a [[Dupin cyclide]], an inversion of the [[torus]]. Thus Soddy's construction shows that a cyclide of Dupin is the envelope of a 1-parameter family of spheres in two different ways, and each sphere in either family is tangent to two spheres in same family and three spheres in the other family.<ref>{{Harvnb|Coxeter|1952}}</ref> This result was probably known to [[Charles Dupin]], who discovered the cyclides that bear his name in his 1803 dissertation under [[Gaspard Monge]].<ref>{{Harvnb|O'Connor|Robertson|2000}}</ref>
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| ==Relation to Steiner chains==
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| [[Image:Steiner chain animation opt.gif|right|thumb|150px|Figure 4: Steiner chain of six circles corresponding to a Soddy's hexlet.]]
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| The intersection of the hexlet with the plane of its spherical centers produces a [[Steiner chain]] of six circles.
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| ==Parabolic and hyperbolic hexlets==
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| It is assumed that spheres A and B are the same size.
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| In any [[ellipse|elliptic]] hexlet, such as the one shown at the top of the article, there are two tangent planes to the hexlet. In order for an elliptic hexlet to exist, the radius of C must be less than one quarter that of A. If C's radius is one quarter of A's, each sphere will become a [[plane (geometry)|plane]] in the journey. The inverted image shows a normal elliptic hexlet, though, and in the [[parabola|parabolic]] hexlet, the point where a sphere turns into a plane is precisely when its inverted image passes through the centre of inversion. In such a hexlet there is only one tangent plane to the hexlet. The line of the centres of a parabolic hexlet is a parabola.
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| If C is even larger than that, a [[hyperbola|hyperbolic]] hexlet is formed, and now there are no tangent planes at all. Label the spheres ''S''<sub>1</sub> to ''S''<sub>6</sub>. ''S''<sub>1</sub> thus cannot go very far until it becomes a plane (where its inverted image passes through the centre 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.
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| The limiting case is when A, B and C are all the same size. The hexlet now becomes straight. ''S''<sub>1</sub> is small as it passes through the hole between A, B and C, and grows till it becomes a plane tangent to them. The centre of inversion is now also with a point of tangency with the image of ''S''<sub>6</sub>, so it is also a plane tangent to A, B and C. As ''S''<sub>1</sub> proceeds, its concavity is reversed and now it surrounds all the other spheres, tangent to A, B, C, ''S''<sub>2</sub> and ''S''<sub>6</sub>. ''S''<sub>2</sub> pushes upwards and grows to become a tangent plane and ''S''<sub>6</sub> shrinks. ''S''<sub>1</sub> then obtains ''S''<sub>6</sub>'s former position as a tangent plane. It then reverses concavity again and passes through the hole again, beginning another round trip. Now the line of centres is a [[degeneracy (mathematics)|degenerate]] hyperbola, where it has collapsed into two straight lines.<ref name="ogilvy"/>
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| ==Sangaku tablets==
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| [[Image:Sangaku of Soddy's hexlet in Samukawa Shrine.jpg|thumb|right|150px|Replica of [[Sangaku]] at Hōtoku museum in [[Samukawa Shrine]].]]
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| The Japanese mathematicians analysed the packing problems in which circles and polygons, balls and polyhedrons come into contact and often found the relevant theorems independently before their discovery by Western mathematicians. The [[Sangaku]] about hexlet was made by Irisawa Shintarō Hiroatsu in the family of Uchida Itsumi and dedicated to [[Samukawa Shrine]] on May, 1822. The original sangaku has been lost and recorded in the Uchida's book of ''Kokinsankagami'' on 1832. The replica of the sangaku was made from the record and dedicated to Hōtoku museum in Samukawa Shrine on August, 2009.<ref>''Dictionary of Wasan'' (''Wasan no Jiten'' in Japanese), p.443</ref>
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| The sangaku by Irisawa consists of 3 problems and the third problem relates to Soddy's hexlet: "the diameter of the outer circumscribing sphere is 30 [[cun (unit)|sun]]. The diameters of the nucleus balls are 10 sun and 6 sun each. The diameter of one of the balls in the chain of balls is 5 sun. Then I asked for the diameters of the remaining balls. The answer is 15 sun, 10 sun, 3.75 sun, 2.5 sun and 2+8/11 sun."<ref>''Sangaku Collection in Kanagawa prefecture'' (''Kanagawa-ken Sangaku-syû'' in Japanese), pp.21-24.</ref>
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| By his answer, the method to calculate the diameters of the balls is written down and can consider it the following formulas to be given in the modern scale. If the ratio of the diameter of the outside ball to the nucleus balls are ''a''<sub>1</sub>, ''a''<sub>2</sub>, and if the ratio of the diameter to the chain balls are ''c''<sub>1</sub>, ..., ''c''<sub>6</sub>. I want to represent c''<sub>2</sub>, ..., ''c''<sub>6</sub> by ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''c''<sub>1</sub>. If
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| :<math>K=\sqrt{3\left( a_1 a_2+a_2 c_1+c_1 a_1- \left( \frac{a_1+a_2+c_1+1}{2} \right)^2 \right)}</math>
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| then,
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| :<math>\begin{align}
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| c_2&=(a_1+a_2+c_1-1)/2-K \\
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| c_3&=(3a_1+3a_2-c_1-3)/2-K \\
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| c_4&=2a_1+2a_2-c_1-2 \\
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| c_5&=(3a_1+3a_2-c_1-3)/2+K \\
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| c_6&=(a_1+a_2+c_1-1)/2+K.
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| \end{align}
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| </math>.
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| Then ''c''<sub>1</sub> + ''c''<sub>4</sub> = ''c''<sub>2</sub> + ''c''<sub>5</sub> = ''c''<sub>3</sub> + ''c''<sub>6</sub>. If ''r''<sub>1</sub>, ..., ''r''<sub>6</sub> are the diameters of six balls, then we get the formula:
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| : <math>\frac{1}{r_1}+\frac{1}{r_4}=\frac{1}{r_2}+\frac{1}{r_5}=\frac{1}{r_3}+\frac{1}{r_6}.</math>
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| ==See also==
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| *[[Descartes' theorem]]
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| *[[Inversive geometry]]
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| ==Notes==
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| {{reflist|1}}
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| ==References==
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| * {{citation | first = Hiroshi | last = Amano | year = 1992 | title = Sangaku Collection in Kanagawa prefecture (Kanagawa-ken Sangaku-syū in Japanese) | publisher = Amano, Hiroshi}}.
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| * {{citation | last = Coxeter | first = HSM | title = Interlocked rings of spheres | journal = [[Scripta Mathematica]] | volume = 18 | year = 1952 | pages = 113–121}}.
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| * {{citation | last1 = Fukagawa | first1 = Hidetoshi | last2 = Rothman | first2 = Tony | year = 2008 | title = Sacred Mathematics: Japanese Temple Geometry | publisher = Princeton University Press | isbn = 978-0-691-12745-3 | url = http://press.princeton.edu/titles/8646.html}}
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| * {{citation| last1=O'Connor|first1= John J.|first2=Edmund F.|last2= Robertson|chapter-url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html|chapter=Pierre Charles François Dupin|title=[[MacTutor History of Mathematics archive]]|year=2000}}.
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| * {{citation | last = Ogilvy | first = C.S. | year = 1990 | title = Excursions in Geometry | publisher = Dover | isbn = 0-486-26530-7}}.
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| * {{citation| last = Soddy | first = Frederick | year = 1937 | title = The bowl of integers and the hexlet | journal = [[Nature (journal)|Nature]]|place= London | pages = 77–79 | volume = 139| doi = 10.1038/139077a0 | issue=3506}}.
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| * {{citation | last = Rothman | first = T | year = 1998 | title = Japanese Temple Geometry | journal = [[Scientific American]] | volume = 278 | pages = 85–91}}.
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| * {{citation | editor = Yamaji, Katsunori; Nishida, Tomomi | year = 2009 | title = Dictionary of Wasan (Wasan no Jiten in Japanese) | publisher = Asakura | isbn = 978-4-254-11122-4}}.
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| ==External links== | |
| *{{mathworld|title=Hexlet|urlname=Hexlet}}
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| *{{cite web|url=http://members.ozemail.com.au/~llan/soddy.html|title=Animation of Soddy's hexlet|author= B. Allanson}}
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| * [http://www.ballstructure.com/Japanese_Math/J_Temple_Geometry.HTM Japanese Temple Geometry] – The animation 0 of SANGAKU PROBLEM 0 shows the case which the radiuses of spheres A and B are equal each other and the centers of spheres A, B and C are on the line. The animation 1 shows the case which the radiuses of spheres A and B are equal each other and the centers of spheres A, B and C are ''not'' on the line. The animation 2 shows the case which the radiuses of spheres A and B are ''not'' equal each other. The animation 3 shows the case which the centers of spheres A, B and C are on the line and the radiuses of spheres A and B are variable.
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| * [http://www.wasan.earth.linkclub.com/kanagawa/samukawa.html Replica of Sangaku at Hōtoku museum in Samukawa Shrine] – The third problem relates to Soddy's hexlet.
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| [[Category:Theorems in geometry]]
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| [[Category:Euclidean solid geometry]]
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