24-cell Viewer
Rotate and slice the 24-cell, the regular 4D polytope with no equivalent in 3D.
This is a free online 24-cell viewer. The 24-cell is a regular four-dimensional polytope made of 24 octahedra, glued face to face, six around each edge. It is the only regular polytope, in any dimension, that is not a version of a shape that exists in all dimensions (simplex, cube, cross-polytope) and that is its own dual, which is why it is often called the most beautiful of the 4D polytopes.
Here you can rotate it through the fourth dimension, watch its shadow, and slice it with our 3D space to see octahedra, cuboctahedra and cubes appear. Download the result as a PNG image, an animated GIF or a WebM video. Everything is drawn by your own graphics card: nothing is uploaded and nothing needs to be installed.
Options
Drag the image to turn the camera around the object, and shift-drag or right-drag to rotate the object through the fourth dimension. Use the mouse wheel to zoom. On a phone, drag with one finger and pinch with two to zoom; to rotate through the fourth dimension, change "Dragging the image".
How to use the 24-cell viewer?
- Start with a preset - "Turning through 4D" shows the slice inside the shadow while the 24-cell rotates in the plane XW. "Shadow" shows its 96 edges in perspective. "Slices, cell first" and "Slices, vertex first" push it through our space in two different orientations.
- Compare the two kinds of slices - Cell first, the slice starts as an octahedron (one of the 24 cells), gets its corners cut into a truncated octahedron, and becomes a cuboctahedron in the middle. Vertex first, it starts as a tiny cube, grows, gets its edges cut off, and becomes a rhombic dodecahedron in the middle.
- Rotate it yourself - Shift-drag or right-drag the image to turn the 24-cell through the fourth dimension. A normal drag turns the camera, and the mouse wheel zooms.
- Count the cells - With "A color for each cell", every face of the slice is a piece of a different octahedral cell.
- Download it - Save the current frame as a PNG image, or one loop of the animation as a GIF or a WebM video.
Settings
Below are the options you can change to explore the object and to create your image or animation.
| Field | Description |
|---|---|
| Show | Slice shows the solid part of the object that is inside our 3D space, like the cross-section of an apple cut by a knife. Shadow shows the whole object, projected from 4D to 3D, as lines. Both shows the slice inside a faint shadow, which tells where the slice is in the object. |
| Position of the slice along W | Moves our 3D space along the fourth axis, W. The objects go from about -1 to 1, so the slice changes shape as it goes through them, and it is empty outside of that range. |
| Projection of the shadow | Perspective makes the parts of the object that are closer to the 4D eye look larger, which gives the famous "cube inside a cube" image of the tesseract. Orthographic simply drops the coordinate W, so the parts that are closer or farther in W have the same size. |
| Distance of the 4D eye | How far from the object the 4D eye is, in the perspective projection. A small distance gives a strong perspective, with a big difference between the near and the far parts. |
| Dragging the image | What happens when you drag the image: turn the camera around the object, as in any 3D viewer, or rotate the object through the fourth dimension, in the plane formed by the direction of the drag and the axis W. Shift-drag or right-drag does the other one. |
| Field of view | The angle seen by the 3D camera, from the top of the image to the bottom. Small angles look flatter, big angles exaggerate the 3D perspective. |
| XY, XZ, YZ, XW, YW and ZW angles | The orientation of the object in 4D. In four dimensions, an object turns in a plane, not around an axis, and there are six planes. XY, XZ and YZ are the ordinary rotations of 3D. XW, YW and ZW mix one of our directions with the fourth one, and turn the object "through" the fourth dimension. |
| Duration of one loop | The time, in seconds, of one loop of the animation, which is also the length of the downloaded GIF or video. |
| Turns in each plane, per loop | How many full turns the object makes in each plane during one loop. Turning in two planes at the same time, like XW and YZ, gives a "double rotation", which only exists in 4D. Whole numbers make an animation that loops perfectly. |
| Move the slice back and forth along W | Moves our 3D space through the object during the loop, so you see the object appear, change shape and disappear, the way a 3D object would look to a 2D creature as it passes through its plane. The distance of the sweep is how far the slice goes on each side. |
| Turns of the camera per loop | Turns the 3D camera around the object during the loop, to show the slice and the shadow from every side. |
| Color of the slice | A different color for each cell (the 3D faces of a 4D polytope), colors that follow the direction of each point in 4D (so they stay attached to the object while it turns), colors that show whether the surface faces the +W side (orange) or the -W side (blue) of our space, or a single color. |
| Edges of the slice | Draws the edges of the polyhedron formed by the slice of a polytope, with the chosen width and color. |
| Color of the shadow | Colors the lines of the shadow by their coordinate W, from blue (toward -W) to orange (toward +W), or with a single color. The lines that are farther from the camera fade into the background, which makes the depth easier to see. |
| Width of the lines / Opacity / Vertices | The width of the lines of the shadow, its opacity when it is drawn with the slice, and whether the vertices of a polytope are drawn as dots. |
| Background colors | The colors at the top and at the bottom of the background, which form a vertical gradient. |
| Samples per pixel | How many rays are calculated for each pixel of the slice, to smooth its edges. While the object moves, the tool may use fewer samples or fewer pixels to keep the animation smooth on slower graphics cards. |
| Resolution / Width / Height | The size of the image in pixels, which is also the size of the PNG download. Choose one of the common resolutions, or type any width and height. |
| Download the animation | Saves one loop of the animation, starting from the current frame, as an animated GIF or a WebM video, with the chosen width and number of frames per second. |
What is the 24-cell?
In 3D there are five regular polyhedra, the Platonic solids. In 4D there are six regular polytopes, and five of them are the 4D versions of the Platonic solids: the 5-cell (tetrahedron), the tesseract (cube), the 16-cell (octahedron), the 120-cell (dodecahedron) and the 600-cell (icosahedron). The sixth one, the 24-cell, is new: it has no 3D version, and in five dimensions and above only three regular polytopes are left.
The 24-cell has 24 vertices, 96 edges, 96 triangular faces and 24 octahedral cells. Its vertices can be placed at all the permutations of (±1, ±1, 0, 0), which are also the centers of the edges of a tesseract, or equivalently at the 8 vertices of a 16-cell plus the 16 vertices of a tesseract. It is self-dual: putting a vertex in the center of each of its cells gives another 24-cell. In 3D, putting a square pyramid of the right height on each face of a cube gives the rhombic dodecahedron; in 4D, putting a cubic pyramid on each of the 8 cells of a tesseract gives the 24-cell. That is why its vertex-first slice is a rhombic dodecahedron in the middle.
The 24-cell also fills 4D space with copies of itself, with no gaps, like cubes fill 3D space. And its 24 vertices are the centers of the 24 spheres that touch a central sphere in the densest known packing of spheres in 4D.
Frequently Asked Questions (FAQ)
Is this 24-cell viewer free?Yes, it is 100% free and works directly in your browser, with no installation and no sign-up required.
Why is the 24-cell special?It is the only regular polytope that has no equivalent in other dimensions, and it is self-dual like the tetrahedron, although its cells are octahedra. It is also the shape of the densest known packing of spheres in 4D, where every sphere touches 24 others.
What is the 24-cell also called?It is also known as the icositetrachoron, the octaplex, the hyperdiamond or the polyoctahedron. "24-cell" simply says that it has 24 cells.
Why is the middle slice a cuboctahedron in one preset and a rhombic dodecahedron in the other?Because our space cuts the 24-cell in two different directions. Cell first, the middle slice contains 12 of the 24 vertices, which are the corners of a cuboctahedron (8 triangles and 6 squares). Vertex first, it contains 6 vertices and crosses 8 edges, which gives the 14 corners of a rhombic dodecahedron (12 rhombi). The two polyhedra are duals of each other, a reflection of the fact that the 24-cell is its own dual.
Can I see other 4D shapes?Yes. The 4D Object Viewer has all the regular 4D polytopes, duoprisms and curved shapes, and there are dedicated pages for the tesseract, the hypersphere, the 120-cell and the 600-cell.
Can I use the animations in my own projects?Yes. You can use the images, GIFs and videos in websites, videos, presentations, lessons and printed material.
Related tools
