4D Euclidean space
News Archive
Mar 2023
This month we present the next Catalan solid in our list: the dual of the truncated octahedron, known as the tetrakis hexahedron:
This is a pretty Catalan solid with 24 isosceles triangles as faces, having a 3 : 3 : 4 edge length ratio. It has 35 edges (24 short, 12 long) and 14 vertices, and is a vertex-first perspective projection of the 24-cell into 3D.
We provide the full algebraic coordinates as usual. Check out the tetrakis hexahedron page!
1 Mar 2023:
The next Catalan Solid, the tetrakis hexahedron, is up!
19 Mar 2023:
Added the triangular bipyramid, the dual of the uniform triangular prism. Not to be confused with the Johnson solid J12, which is also a triangular bipyramid, but with different proportions.
Feb 2023
This month we continue with the next 3D Catalan solid, the rhombic dodecahedron:
This polyhedron has been referred to since the earliest days of this website, as the vertex-first projection envelope of the tesseract and the 24-cell. Now, finally, it has its own page!
As usual, we provide full algebraic coordinates, and, continuing the tradition of last month's Catalan solid, include an animation of a rotating rhombic dodecahedron. Check out the rhombic dodecahedron page!
1 Feb 2023:
The next Catalan Solid, the rhombic dodecahedron, is up!
2 Feb 2023: some exciting site changes are happening!
Experimental new feature: this site now supports dark mode! Enable dark mode in your browser or device settings and reload to see it. You may need to clear your browser cache for some of the changes to take effect. Please let us know if there are any issues, such as missing images or images that are hard to see.
Replace animated GIFs with APNGs. Browsers that don't support APNG should still see the old GIFs. If not, please let us know!
General site cleanup: fix invalid markup, repair broken links.
Jan 2023
This year we will begin to present the Catalan solids, the duals of the uniform polyhedra. The first and simplest among them is the triakis tetrahedron, a face-transitive polyhedron bounded by isosceles triangles with edge ratio 3 : 3 : 5:
As usual, we provide the full algebraic coordinates. There's even an animation that gives you a better feel of its overall shape.
Over the next few months we will present the 3D Catalan solids as well as their generalizations, the bipyramids and polygonal trapezohedra, the duals of the prisms and antiprisms. Then we will move on to the 4D Catalan solids, the duals of the 4D uniform polychora. Stay tuned!
6 Jan 2023:
We officially begin the Catalan project: a project to cover all the 3D Catalan solids, the duals of prisms and antiprisms, and then the 4D Catalan solids!
11 Jan 2023:
Standardized the coordinates of several uniform polyhedra in the dodecahedron family to have a consistent orientation with each other.
Standardized the coordinates of the snub dodecahedron to length 2. There seems to be no nice algebraic presentation for them; so we also provide accurate numerical coordinates up to 15 digits.
Also tweaked the stylesheet for a few pages. If you notice anything odd or out of place, please let us know!
28 Jan 2023:
Added a family portrait of the Catalan solids, including the duals of some prisms and antiprisms.
Added also the pentagonal hexecontahedron as background image to the Catalan solids page.
