5-cell

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C 5 , hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex (Coxeter's α 4 polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides. The regular 5-cell is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes (the four-dimensional analogues of the Platonic solids). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another. No solution exists in three dimensions.

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C 5 , hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex (Coxeter's α 4 polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides. The regular 5-cell is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes (the four-dimensional analogues of the Platonic solids). A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another. No solution exists in three dimensions. File:Simple 4D rotation of a 5-cell, in X-Y plane.webm|loop|Simply rotating in X-Y plane File:Simple 4D rotation of a 5-cell, in Z-W plane.webm|loop|Simply rotating in Z-W plane File:Double 4D rotation of a 5-cell.webm|loop|Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio File:Isoclinic left 4D rotation of a 5-cell.webm|loop|Left isoclinic rotation File:Isoclinic right 4D rotation of a 5-cell.webm|loop|Right isoclinic rotation (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (\phi, \phi, \phi, \phi), with edge length 2 \sqrt{2} , where \phi is the golden ratio. While these coordinates are not origin-centered, subtracting (1,1,1,1)/\left(2-\tfrac{1}{\phi}\right) from each translates the 4-polytope's circumcenter to the origin with radius 2\left(\phi-1/\left(2-\tfrac{1}{\phi}\right)\right) =\sqrt{\tfrac{16}{5}}\approx 1.7888 , with the following coordinates: \begin{align} \left(\tfrac{2}{\phi}-3, 1, 1, 1\right)/\left(\tfrac{1}{\phi}-2\right), &\quad \left(1,\tfrac{2}{\phi}-3,1,1 \right)/\left(\tfrac{1}{\phi}-2\right) \\ \left(1,1,\tfrac{2}{\phi}-3,1 \right)/\left(\tfrac{1}{\phi}-2\right) &\quad \left(1,1,1,\tfrac{2}{\phi}-3 \right)/\left(\tfrac{1}{\phi}-2\right) \\ \left(\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi},\tfrac{2}{\phi} \right)/\left(\tfrac{1}{\phi}-2\right) \end{align} The following set of origin-centered coordinates with the same radius and edge length as above can be seen as a hyperpyramid with a regular tetrahedral base in 3-space: \left( 1, 1, 1, \frac{-1}\sqrt{5}\right) \left( 1,-1,-1,\frac{-1}\sqrt{5} \right) \left(-1, 1,-1,\frac{-1}\sqrt{5} \right) \left(-1,-1, 1,\frac{-1}\sqrt{5} \right) \left( 0, 0, 0,\frac{4}\sqrt{5} \right) Scaling these or the previous set of coordinates by \tfrac{\sqrt{5}}{4} give unit-radius origin-centered regular 5-cells with edge lengths \sqrt{\tfrac{5}{2}} . The hyperpyramid has coordinates: \left( \sqrt{5}, \sqrt{5}, \sqrt{5}, -1 \right)/4 \left( \sqrt{5},-\sqrt{5},-\sqrt{5}, -1 \right)/4 \left(-\sqrt{5}, \sqrt{5},-\sqrt{5}, -1 \right)/4 \left(-\sqrt{5},-\sqrt{5}, \sqrt{5}, -1 \right)/4 \left( 0, 0, 0, 1 \right) Coordinates for the vertices of another origin-centered regular 5-cell with edge length 2 and radius \sqrt{\tfrac{8}{5}}\approx 1.265 are: \left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right) \left( \frac{1}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right) \left( \frac{1}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right) \left( -2\sqrt{\frac{2}{5}},\ 0,\ 0,\ 0 \right) Scaling these by \sqrt{\tfrac{5}{8}} to unit-radius and edge length \sqrt{\tfrac{5}{2}} gives: \left(\sqrt{3}, \sqrt{5}, \sqrt{10},\pm\sqrt{30} \right)/(4\sqrt{3}) \left(\sqrt{3}, \sqrt{5}, -\sqrt{40},0\right)/(4\sqrt{3}) \left(\sqrt{3},-\sqrt{45},0,0\right)/(4\sqrt{3}) \left(-1, 0, 0, 0 \right) The vertices of a 4-simplex (with edge and radius 1) can be more simply constructed on a hyperplane in 5-space, as (distinct) permutations of (0,0,0,0,1) or (0,1,1,1,1); in these positions it is a facet of, respectively, the 5-orthoplex or the rectified penteract. The compound of two 5-cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and blue 5-cell vertices and edges. This compound has 3,3,3 symmetry, order 240. The intersection of these two 5-cells is a uniform bitruncated 5-cell. = ∩ . This compound can be seen as the 4D analogue of the 2D hexagram { } and the 3D compound of two tetrahedra.

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