# Bixylodiminished hydrochoron (EntityTopic, 13)

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- | The '''bixylodiminished hydrochoron''', abbreviated '''BXD''', is a [[CRF polychoron]] obtained by removing the vertices of two of the five inscribed [[xylochora]] from a [[hydrochoron]]. Removing the vertices of only one yields the [[snub demitesseract]], which can therefore also be called the (mono) ''xylodiminished hydrochoron''. Like the snub demitesseract, it exists in two chiral forms. Surprisingly, although the resulting polychoron is not [[uniform]], it is both cell- and vertex-transitive; its 48 cells are [[tridiminished icosahedra]]. This cell transitivity gives it a much simpler [[FLD]], with only four nodes, than the vast majority of other CRF polychora and even many uniform polychora. Finally, if the vertices of a third xylochoron are removed, this gives the (non-CRF) dual of the BXD, the [[trixylodiminished hydrochoron]]. | + | The '''bixylodiminished hydrochoron''', abbreviated '''BXD''', is a [[CRF polychoron]] obtained by removing the vertices of two of the five inscribed [[xylochora]] from a [[hydrochoron]]. Removing the vertices of only one yields the [[snub demitesseract]], which can therefore also be called the (mono) ''xylodiminished hydrochoron''. Like the snub demitesseract, it exists in two chiral forms. Surprisingly, although the resulting polychoron is not [[uniform]], it is both cell- and vertex-transitive with sides of equal length; its 48 cells are [[tridiminished icosahedra]]. This cell transitivity gives it a much simpler [[FLD]], with only four nodes, than the vast majority of other CRF polychora and even many uniform polychora. Finally, if the vertices of a third xylochoron are removed, this gives the (non-CRF) dual of the BXD, the [[trixylodiminished hydrochoron]]. |

While the tridiminished icosahedron has four types of edges (top to pen, pen to pen, pen to lat, lat to bot), the BXD has only two types: ''pen-3-lat'', which has four cells around each instance, separated by three pentagons and one lateral, and the ''top-pen-lat-top loop'', which has three cells around each instance and includes the other laterals. Note that the six pen-to-lat edges are divided into two groups of three due to the chirality of the BXD, the first group ending up as pen-3-lats, the second ending up as loops. From this information and the fact that it has 48 cells, we can work out that there are 6×48÷4 = 72 edges of the pen-3-lat type and 9×48÷3 = 144 edges of the loop type. | While the tridiminished icosahedron has four types of edges (top to pen, pen to pen, pen to lat, lat to bot), the BXD has only two types: ''pen-3-lat'', which has four cells around each instance, separated by three pentagons and one lateral, and the ''top-pen-lat-top loop'', which has three cells around each instance and includes the other laterals. Note that the six pen-to-lat edges are divided into two groups of three due to the chirality of the BXD, the first group ending up as pen-3-lats, the second ending up as loops. From this information and the fact that it has 48 cells, we can work out that there are 6×48÷4 = 72 edges of the pen-3-lat type and 9×48÷3 = 144 edges of the loop type. |

## Latest revision as of 17:18, 3 February 2018

The **bixylodiminished hydrochoron**, abbreviated **BXD**, is a CRF polychoron obtained by removing the vertices of two of the five inscribed xylochora from a hydrochoron. Removing the vertices of only one yields the snub demitesseract, which can therefore also be called the (mono) *xylodiminished hydrochoron*. Like the snub demitesseract, it exists in two chiral forms. Surprisingly, although the resulting polychoron is not uniform, it is both cell- and vertex-transitive with sides of equal length; its 48 cells are tridiminished icosahedra. This cell transitivity gives it a much simpler FLD, with only four nodes, than the vast majority of other CRF polychora and even many uniform polychora. Finally, if the vertices of a third xylochoron are removed, this gives the (non-CRF) dual of the BXD, the trixylodiminished hydrochoron.

While the tridiminished icosahedron has four types of edges (top to pen, pen to pen, pen to lat, lat to bot), the BXD has only two types: *pen-3-lat*, which has four cells around each instance, separated by three pentagons and one lateral, and the *top-pen-lat-top loop*, which has three cells around each instance and includes the other laterals. Note that the six pen-to-lat edges are divided into two groups of three due to the chirality of the BXD, the first group ending up as pen-3-lats, the second ending up as loops. From this information and the fact that it has 48 cells, we can work out that there are 6×48÷4 = 72 edges of the pen-3-lat type and 9×48÷3 = 144 edges of the loop type.

## Equations

- The hypervolumes of a bixylodiminished hydrochoron with side length
*l*are given by:

total edge length = 216l

total surface area = 6(3√(25+10√5) + 5√3) ·l^{2}

surcell volume = 2(15+7√5) ·l^{3}

bulk =Unknown

## Incidence matrix

Dual: trixylodiminished hydrochoron

# | TXID | Va | Ea | Eb | 3a | 3b | 5a | C1a | Type | Name |
---|---|---|---|---|---|---|---|---|---|---|

0 | Va | = point | ; | |||||||

1 | Ea | 2 | = digon | ; loop | ||||||

2 | Eb | 2 | = digon | ; pen-3-lat | ||||||

3 | 3a | 3 | 3 | 0 | = triangle | ; verticals | ||||

4 | 3b | 3 | 2 | 1 | = triangle | ; laterals | ||||

5 | 5a | 5 | 2 | 3 | = pentagon | ; pentagons | ||||

6 | C1a | 9 | 9 | 6 | 2 | 3 | 3 | = tridiminished icosahedron | ; | |

7 | H4.1a | 72 | 144 | 72 | 48 | 72 | 72 | 48 | = bixylodiminished hydrochoron |
; |

## Usage as facets

*This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.*

## Software models

## External links

Notable Tetrashapes
| |

Regular:
| pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |

Powertopes:
| triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |

Circular:
| glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |

Torii:
| tiger • torisphere • spheritorus • torinder • ditorus |