120-cell Viewer
Rotate the 120-cell, the 4D polytope made of 120 dodecahedra, and see its shadow and its slices.
This is a free online 120-cell viewer. The 120-cell is the four-dimensional version of the dodecahedron: a regular polytope made of 120 dodecahedra, with 600 vertices, 1200 edges and 720 pentagonal faces. It is the largest and most complex of the six regular 4D polytopes, and its shadow is a dense lace of pentagons.
Here you can rotate it through the fourth dimension, watch its shadow, and slice it with our 3D space to see a dodecahedron grow into intricate polyhedra. 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 120-cell viewer?
- Start with a preset - "Turning through 4D" shows the 1200 edges of the shadow in perspective while the 120-cell rotates in the plane XW. "Slices" moves our 3D space through it, and "Slice and shadow" shows both.
- Watch the first cell - In "Slices", the 120-cell enters our space cell first, so the slice is one of its 120 dodecahedra. It stays a dodecahedron for a while, because the cells around the first one are tilted away from our space, and then the neighbor cells start to show up as new faces.
- Look at the middle - At W = 0, the slice is a polyhedron with 80 vertices, 12 pentagons and 30 hexagons: every face is a piece of a different dodecahedral cell.
- Rotate it yourself - Shift-drag or right-drag the image to turn the 120-cell through the fourth dimension. A normal drag turns the camera, and the mouse wheel zooms. Thinner lines and fewer samples make the shadow easier to read and faster to draw.
- 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 120-cell?
The dodecahedron is the Platonic solid made of 12 pentagons, three around each vertex. The 120-cell is its 4D version: 120 dodecahedra, three around each edge and four around each vertex, closing up into a finite polytope. Its 600 vertices all lie on a hypersphere, and its 1200 edges, 720 pentagons and 120 cells are all identical.
It is the dual of the 600-cell: putting a vertex in the center of each of the 600 tetrahedra of the 600-cell gives the 600 vertices of the 120-cell, and the centers of the 120 dodecahedra of the 120-cell are the 120 vertices of the 600-cell. This tool builds the 120-cell exactly that way. The two polytopes share their symmetry group, which has 14,400 elements, the largest of all the regular polytopes in 4D.
Ludwig Schläfli discovered the 120-cell around 1852, together with the other regular 4D polytopes. Its perspective shadow, with a small dodecahedron at the center surrounded by layers of distorted dodecahedra, is one of the most famous images of four-dimensional geometry.
Frequently Asked Questions (FAQ)
Is this 120-cell viewer free?Yes, it is 100% free and works directly in your browser, with no installation and no sign-up required.
What is the 120-cell also called?It is also known as the hecatonicosachoron, the dodecacontachoron or the hyperdodecahedron. "120-cell" simply says that it has 120 cells.
Why is the shadow so hard to read?Because the 1200 edges of the 120-cell are all squeezed into 3D, where they overlap. Lower the "Width of the lines", use the orthographic projection, or color the shadow by the coordinate W: the farther lines are also dimmer, which separates the layers. The slice is often easier to understand, because it is a single solid polyhedron.
Why is the slice a dodecahedron for so long?The 120-cell enters our space with one of its cells first. Its neighbor cells meet that first cell at an angle, so our space must move a bit farther along W before it reaches them. Until then, the slice is a copy of the first cell, a dodecahedron.
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 24-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.
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