4D Euclidean space
News Archive
Dec 2023
This month our polytope of the month is the innocent-looking dual of the uniform pentagonal prism, the pentagonal bipyramid:
While topologically equivalent, this polyhedron should not be confused with the Johnson solid J13, which is also a pentagonal bipyramid, but with different proportions. While J13's height is about 1.05 of its edge length, the height of this pentagonal bipyramid is about 2.34 of the length of its equatorial edges (short edges). Whilst the faces of J13 are equilateral triangles, the faces of this polyhedron are isosceles triangles with an edge length ratio of 3−φ : 2, or approximately 0.691 : 1.
The reason this polyhedron is interesting is because it occurs in a 4D Catalan polychoron, which will be next month's feature. Stay tuned!
In the meantime, check out the pentagonal bipyramid page!
3 Dec 2023:
The Polytope of the Month for December is up!
Nov 2023
This month, we return to 4D with another 4D Catalan polytope:
This is the square antitegmatic hecatontetracontatetrachoron, a beautiful polychoron whose dual is the runcinated 24-cell. It sports 144 square trapezohedra as cells, 576 kite-shaped polygons, 672 edges, and 240 vertices.
Find out more on the square antitegmatic hecatontetracontatetrachoron page! As usual, the full algebraic coordinates are given.
As a bonus, we also included an alternative analysis of the structure of this polychoron by its correspondence to the Hopf fibration of the 3-sphere. Check it out!
3 Nov 2023:
The Polytope of the Month for November is now up!
Oct 2023
This month, we introduce the square trapezohedron:
It is the dual of the square antiprism, and is among the class of duals of uniform polyhedra that includes the Catalan solids. It is bounded by 8 kite-shaped faces, 16 edges (8 long, 8 short), and 10 vertices. Like the square antiprism, its proportions are closely related to ∜2.
Find out more on the square trapezohedron page! As usual, we provide full algebraic coordinates.
1 Oct 2023:
The Polytope of the Month for October is now up!
