# -: J :-

j, jn *
When j is followed by a value, it refers to the solution of the isomorphic root or isoroot j(n) = √(n+2)+√(n-2))/2. The following table shows the more important geometric isoroots.
- twelfty decimal symbol
j3 1:7419 8287,V8V3 43E0 1.618033988749894848 φ, τ
j4 1:E198 7978,8151 V4E3 1.931851652578136573 ω
j6 2:4984 8104,3529 0779 2.414213562373095048 α
j103:1766 2499,9016 9728 3.146264369941972342 β, √3+√2
Johnson
This usually refers to Professor Norman Johnson.
Johnson Polyhedra, any of the 92 convex polyhedra formed from regular polygons.
Johnson Notation
A notation based on naming the nodes of the Wythoff graph, according to the sequence [ ] truncate, cantelate, runcinate. That is, the nodes of a polychoron make xtcr. The names follow Bower's naming, but the order is slightly different.
 ``` Bowers Johnson -------------- ------------------- oxo xxo truncate truncate xox rhombi- rhombi- xxx rhombitruncate truncated oxoo rectate rectate xxoo truncated truncated oxxo bitruncated xoxo rhombi cantelated xxxo rhombitruncate cantetruncate xoox prismato runcinate xxox prismatorhombi runcitruncate xxxx prismatorhombitru omnitruncate soxo cantuisnub soox runcisnub soxx runcicantisnub middle form, after Kepler's 'Cuboctahedron' Dual Jonathan names xoxx, not xxox. ```
join
To place two polytopes together, such that they share common surtopes. See also mount.
join Bowers
Jonathan Bowers proposed a form of join, where when two polytopes are placed together, the common face is removed to make a larger one.
join*
Conway's name for a product that corresponds to the tegum and pyramid product combined. The notion is that one forms a union of vertices in orthogonal bases, and throws a skin over the result.
In a complete join has the centres of polytopes match, and thus gives to the tegum product.
An incomplete join has the centres of the bases at different places, and thus corresponds to a pyramid product.
The Conway-operator corresponds to a surtegmate on the edges of a polytope. The operator is dual to the ambo operator.