Bitruncated 24-cell

In geometry, the bitruncated 24-cell is a 4-dimensional uniform polytope (or uniform polychoron) derived from the 24-cell. It is constructed by bitruncating the 24-cell (truncating at halfway to the depth which would yield the dual 24-cell).

Being a uniform polychoron, it is vertex-transitive. In addition, it is cell-transitive, consisting of 48 truncated cubes, and also edge-transitive, with 3 truncated cubes cells per edge and with one triangle and two octagons around each edge.

The 48 cells of the bitruncated 24-cell correspond with the 24 cells and 24 vertices of the 24-cell. As such, the centers of the 48 cells form the root system of type F₄.

Its vertex figure is a "tetragonal disphenoid", a tetrahedron with 2 opposite edges length 1 and all 4 lateral edges length sqrt(2+sqrt(2)).

tructure

The truncated cubes are joined to each other via their octagonal faces in "anti" orientation; i. e., two adjoining truncated cubes are rotated 45 degrees relative to each other so that no two triangular faces share an edge.

The sequence of truncated cubes joined to each other via opposite octagonal faces form a cycle of 8. Each truncated cube belongs to 3 such cycles. On the other hand, the sequence of truncated cubes joined to each other via opposite triangular faces form a cycle of 6. Each truncated cube belongs to 4 such cycles.

Projections

Projection to 2 dimensions

The orthogonal cell-first projection of the bitruncated 24-cell into 2 dimensions has an octagonal envelope. Under this projection, the octagonal faces project onto a central octagon and 8 surrounding octagons (see figure above table).

Projection to 3 dimensions

Orthographic

The following animation shows the orthographic projection of the bitruncated 24-cell into 3 dimensions. The animation itself is a perspective projection from the static 3D image into 2D, with rotation added to make its structure more apparent.

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The images of the 48 truncated cubes are laid out as follows:
* The central truncated cube is the cell closest to the 4D viewpoint, highlighted to make it easier to see. To reduce visual clutter, the vertices and edges that lie on this central truncated cube have been omitted.
* Surrounding this central truncated cube are 6 truncated cubes attached via the octagonal faces, and 8 truncated cubes attached via the triangular faces. These cells have been made transparent so that the central cell is visible.
* The 6 outer square faces of the projection envelope are the images of another 6 truncated cubes, and the 12 oblong octagonal faces of the projection envelope are the images of yet another 12 truncated cubes.
* The remaining cells have been culled because they lie on the far side the bitruncated 24-cell, and are obscured from the 4D viewpoint. These include the antipodal truncated cube, which would have projected to the same volume as the highlighted truncated cube, with 6 other truncated cubes surrounding it attached via octagonal faces, and 8 other truncated cubes surrounding it attached via triangular faces.

Perspective

The following animation shows the cell-first perspective projection of the bitruncated 24-cell into 3 dimensions. Its structure is the same as the previous animation, except that there is some foreshortening due to the perspective projection.

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See also

* 24-cell
* Bitruncated 5-cell, the other cell-transitive, non-regular uniform polychoron
* Uniform polychoron
* Polychoron

External links

* [http://members.aol.com/Polycell/section3.html 3. Convex uniform polychora based on the icositetrachoron (24-cell)] - Model 27