Fully independend of this more than 5 years old thread, this weekend I investigated the 4D equivalents of the rhombohedra myself.

The rhombohedra are obtained when reusing the vertices of any regular polyhedron, but replacing each edge by a rhomb, which by itself is situated tangentially to the former edge center sphere. These rhombs then add further vertices somewhere atop the former faces. This is why those rhombohedra also could be obtained as the convex hull of the compound of dual pairs of polyhedra, provided that the dual polyhedron would be scaled accordingly, that is, when the edges of both polyhedra are taken tangential to a common sphere.

Now let us take over this constructional device to the next dimension, and look what we would get there. Again we start with some regular polychoron. We maintain its vertex set and will use the edges as constructive device. Thus we are looking again for the edge center glome and will have to scale the dual polychoron such that its 2D face center glome would coincide. Then the hulls of the former edge ends and the orthogonal 2D faces each would define some bipyramids, as the counterparts of the rhombs of the 3D setup. (Examples thereof can be seen quite well in Quickfur's partial pic.)

The result then for either pairing of dual polychora will be two polychora (one using the edges of P, one using the edges of P*), fully bounded by these bipyramids only. As we have 6 (convex) regular polychora, we likewise look forward for 6 bipyramidally bounded polychora, aka bipyramidochora.

The edge glome radius R_{} obviously is related to the circumradius R_0 according to R_{}^2 = R_0^2 - r_{}^2 = R_0^2 - (1/2)^2.

The 2D face radius R_{p} of the polychoron {p,q,r} is related to the circumradius R_0 according to R_{p}^2 = R_0^2 - r_{p}^2, where r_{p} then is the 2D circumradius of the polygon {p}.

Therefore, when those glomes ought coincide and the edge size of the starting polychoron P would be unity, then the edge size of the dual polychoron P* would equate to x' = 1 * R_{} / R*_{p*}. Here x' also would be the equatorial edge size of the bipyramids, provided the axial hight would be unity. Accordingly the corresponding lacing edge sizes (within that scaling) then could be calculated as

x"^2 = (x' * r*_{p*})^2 + r_{}^2 = (R_{} * r*_{p*} / R*_{p*})^2 + (1/2)^2.

Of special interest then would be those cases, where one gets x' = x"; for then that bipyramidochoron has a single edge size only, and therefore the used lacing triangles become regular in addition. That is, that bipyramidochoron would become a CRF. For that to take place we have to equate 4 * R_{}^2 * r*_{p*}^2 + R*_{p*}^2 = 4 * R_{}^2 or equivalently r*_{p*}^2 + R*_{p*}^2 / (2 * R_{})^2 = 1.

The following table provides these respective radii for the regular polychora, always given in respective edge units of the starting figures. (Make sure to use the scroll bars!)

- Code: Select all
`-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+`

polychoron {p,q,r} | pen = {3,3,3} | hex = {3,3,4} | tes = {4,3,3} | ico = {3,4,3} |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

R_0 | sqrt(2/5) = 0.632456 | 1/sqrt(2) = 0.707107 | 1 | 1 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

r_{} | 1/2 | 1/2 | 1/2 | 1/2 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

R_{} | sqrt(3/20) = 0.387298 | 1/2 | sqrt(3)/2 = 0.866025 | sqrt(3)/2 = 0.866025 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

r_{p} | 1/sqrt(3) = 0.577350 | 1/sqrt(3) = 0.577350 | 1/sqrt(2) = 0.707107 | 1/sqrt(3) = 0.577350 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

R_{p} | 1/sqrt(15) = 0.258199 | 1/sqrt(6) = 0.408248 | 1/sqrt(2) = 0.707107 | sqrt(2/3) = 0.816497 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

r_{p,q} | sqrt(3/8) = 0.612372 | sqrt(3/8) = 0.612372 | sqrt(3)/2 = 0.866025 | 1/sqrt(2) = 0.707107 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

R_{p,q} | 1/sqrt(40) = 0.158114 | 1/sqrt(8) = 0.353553 | 1/2 | 1/sqrt(2) = 0.707107 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

x' = R_{} / R*_{p*} | 3/2 | 1/sqrt(2) = 0.707107 | 3/sqrt(2) = 2.121320 | 3/sqrt(8) = 1.060660 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

x" = sqrt[(x' * r*_{p*})^2 + r_{}^2] | 1 | 1/sqrt(2) = 0.707107 | sqrt(7)/2 = 1.322876 | sqrt(5/8) = 0.790569 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

y = x"/x' | 2/3 | 1 | sqrt(7/18) = 0.623610 | sqrt(5)/3 = 0.745356 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

#{} | 10 | 24 | 32 | 96 |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

cells | oxo3ooo&#y | oxo4ooo&#x = oct | oxo3oo&#y | oxo3oo&#y |

-------------------------------------+-----------------------+----------------------+-----------------------+----------------------+

- Code: Select all
`-------------------------------------+------------------------------------+-------------------------------------+`

polychoron {p,q,r} | ex = {3,3,5} | hi = {5,3,3} |

-------------------------------------+------------------------------------+-------------------------------------+

R_0 | (1+sqrt(5))/2 = 1.618034 | sqrt[7+3 sqrt(5)] = 3.702459 |

-------------------------------------+------------------------------------+-------------------------------------+

r_{} | 1/2 | 1/2 |

-------------------------------------+------------------------------------+-------------------------------------+

R_{} | sqrt[5+2 sqrt(5)]/2 = 1.538842 | sqrt[27+12 sqrt(5)]/2 = 3.668542 |

-------------------------------------+------------------------------------+-------------------------------------+

r_{p} | 1/sqrt(3) = 0.577350 | sqrt[(5+sqrt(5))/10] = 0.850651 |

-------------------------------------+------------------------------------+-------------------------------------+

R_{p} | sqrt[(7+3 sqrt(5))/6] = 1.511523 | sqrt[(65+29 sqrt(5))/10] = 3.603415 |

-------------------------------------+------------------------------------+-------------------------------------+

r_{p,q} | sqrt(3/8) = 0.612372 | sqrt[(9+3 sqrt(5))/8] = 1.401259 |

-------------------------------------+------------------------------------+-------------------------------------+

R_{p,q} | sqrt[(9+4 sqrt(5))/8] = 1.497676 | sqrt[(47+21 sqrt(5))/8] = 3.427051 |

-------------------------------------+------------------------------------+-------------------------------------+

x' = R_{} / R*_{p*} | sqrt[(35-15 sqrt(5))/8] = 0.427051 | 3 (1+sqrt(5))/4 = 2.427051 |

-------------------------------------+------------------------------------+-------------------------------------+

x" = sqrt[(x' * r*_{p*})^2 + r_{}^2] | (sqrt(5)-1)/2 = 0.618034 | sqrt[(11+3 sqrt(5))/8] = 1.487792 |

-------------------------------------+------------------------------------+-------------------------------------+

y = x"/x' | sqrt[(6+2 sqrt(5))/5] = 1.447214 | sqrt[(9-sqrt(5))/18] = 0.613004 |

-------------------------------------+------------------------------------+-------------------------------------+

#{} | 720 | 1200 |

-------------------------------------+------------------------------------+-------------------------------------+

cells | oxo5ooo&#y | oxo3oo&#y |

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Thus, that bipyramidochoron, which results from hex, is nothing but ico. All other ones exist too, but would not result in CRFs. In fact, all trigonal ones will use oblate bipyramides (x > y), only the pentagonal one uses prolate bipyramids (x < y).

I've to admit, that I mainly was looking for CRFs in the run of those constructions. Sadly this research came out negative. But on the other hand, this thread clearly is right. All these are nothing but some of the 4D Catalans. In fact those which Wendy would write oPmQoRo. (Just as the 3 well-known rhombohedra themselves are nothing but oPmQo.)

--- rk