page_title: Soddy's hexlet
section_title: Sangaku tablets
hierarchical_section_title: Soddy's hexlet / Sangaku tablets
caption_attribution_description: English: Sangaku of Soddy' hexlet at Hôtoku museum in Samukawa Shrine 日本語: 寒川神社の方徳資料館に奉納されているソディの6球連鎖の算額
context_page_description: In geometry, Soddy's hexlet is a chain of six spheres, each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, the three spheres are the blue outer sphere and two spheres above and below the plane the centers of the hexlet spheres lie on. In addition, the hexlet spheres are tangent to a fourth sphere, which is not tangent to the three others.
  According to a theorem published by Frederick Soddy in 1937, 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; in this, Soddy's hexlet is the spherical analog of a Steiner chain of six circles. 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 Kanagawa prefecture.
context_section_description: Japanese mathematicians discovered the same hexlet over one hundred years before Soddy did. They 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. They often published these as sangaku. The sangaku about the hexlet was made by Irisawa Shintarō Hiroatsu in the school of Uchida Itsumi, and dedicated to the Samukawa Shrine in May 1822. The original sangaku has been lost but was recorded in Uchida's book of Kokonsankan in 1832. A replica of the sangaku was made from the record and dedicated to the Hōtoku museum in the Samukawa Shrine in August, 2009.
  The sangaku by Irisawa consists of three problems. The third problem relates to 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 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."
  In his answer, the method for calculating the diameters of the balls is written down and can consider it the following formulas to be given in the modern scale. If the ratios of the diameter of the outside ball to each of the nucleus balls are a₁, a₂, and if the ratios of the diameter to the chain balls are c₁, ..., c₆. we want to represent c₂, ..., c₆ in terms of  a₁, a₂, and c₁. If
  then,
  .
  Then c₁ + c₄ = c₂ + c₅ = c₃ + c₆.
  If  r₁, ..., r₆ are the diameters of six balls, we get the formula:
caption: Replica of Sangaku at Hōtoku museum in Samukawa Shrine.
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