Hypersphere Viewer (3-Sphere)
Watch a 4D hypersphere pass through our 3D space, and see its Hopf fibers in perspective.
This is a free online hypersphere viewer. A hypersphere is the set of points at the same distance from a center in four dimensions. When it passes through our 3D space, we see a sphere that appears as a point, grows, shrinks and disappears, exactly like a ball passing through a sheet of paper looks to a flat creature living in the sheet: a dot, a growing circle, a shrinking circle, and nothing.
You can also see its shadow, drawn with the circles of the Hopf fibration: circles that fill the whole hypersphere, never cross each other, and are linked like the rings of a chain. Download the result as a PNG image, an animated GIF or a WebM video. Everything is calculated 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 hypersphere viewer?
- Watch it pass through - The "Passing through our space" preset moves our 3D space along the fourth axis, W, from one side of the hypersphere to the other. The slice is a sphere whose radius follows the shape of a circle: √(1 - w²).
- See the colors move - With the colors "By direction in 4D", each point of the hypersphere has a color that depends on where it is in 4D. As the slice moves, different parts of the hypersphere cross our space, so the colors of the sphere change although its shape doesn't.
- Look at the Hopf fibers - The "Hopf fibers" preset shows the shadow of the hypersphere drawn with great circles. Turn the object through the fourth dimension with a shift-drag or a right-drag, and watch the circles stay linked.
- Combine both - "Slice and shadow" shows the sphere of the slice inside the shadow, which makes it clear which part of the hypersphere you are looking at.
- 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 a hypersphere?
A circle is all the points of a plane at the same distance from a center, and a sphere is all the points of space at the same distance from a center. The hypersphere is the same idea in four dimensions: all the points (x, y, z, w) with x² + y² + z² + w² = r². Mathematicians call it the 3-sphere, because its surface is three-dimensional, like the surface of an ordinary sphere is two-dimensional (the 2-sphere).
Every slice of a hypersphere is a sphere, and the slices of a hypersphere of radius 1 have the radius √(1 - w²): 1 in the middle, and 0 at w = ±1. This is why it looks like a sphere that inflates and deflates as it passes through our space. Turning the hypersphere changes nothing, because it is perfectly round in every direction; only its colors move.
The surface of the hypersphere can be filled completely with circles that never touch each other. This arrangement, the Hopf fibration, was discovered by Heinz Hopf in 1931. Any two of its circles are linked, like two rings of a chain, and in the shadow they form nested tori of interlocking rings. The circle in the plane XY and the circle in the plane ZW are two of them, and every other circle winds around both.
Frequently Asked Questions (FAQ)
Is this hypersphere viewer free?Yes, it is 100% free and works directly in your browser, with no installation and no sign-up required.
Is the hypersphere the same as the 3-sphere?Yes, in most contexts. "Hypersphere" can mean a sphere in any number of dimensions, and the 3-sphere is the one that lives in 4D space. Its surface has three dimensions, which gives it its name.
Why does the hypersphere look like an ordinary sphere?Because we only see its slice, and every slice of a hypersphere is a sphere. What reveals the fourth dimension is the way the sphere grows and shrinks as the slice moves, and the way the colors change: different parts of the hypersphere cross our space at different positions of the slice.
What would a hypersphere look like if it passed through our world?Exactly like in this tool: a point would appear out of nowhere, grow into a sphere, reach its full size, shrink and vanish. A creature living in a plane would see the same thing with an ordinary ball passing through its plane, with a circle instead of a sphere.
Can I see other 4D shapes?Yes. The 4D Object Viewer has the regular 4D polytopes, duoprisms, the duocylinder and 4D tori, and there are dedicated pages for the tesseract, the 24-cell, 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.
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