Visualizing higher dimensional spheres

• Hornbein
In summary: The 3-sphere can be parameterized with polar coordinates as sin(x)e^it, cos(x)e^iu, with 0<=x<pi/2, -pi<=t,u<pi. Map to the three axis of R^3 using x, sin(x)t, and cos(x)u. In summary, a higher dimensional sphere seems impossible to me. The best that can be done is to come up with several different representations, each of which is inaccurate in a different way.
Hornbein
Visualizing a higher-dimensional sphere seems impossible to me. The best that can be done is to come up with several different representations, each of which is inaccurate in a different way.

A property of n-spheres is that they have n/2 planes of rotation, each at right angles to the other. Amazingly even in a rigid rotation the planes can have completely different periods of rotation.

The easiest case is the 3-sphere, which is embedded in Euclidian four-space. To simplify things, let's set the two periods of rotation to be equal. Instead of a plane of rotation we can consider a circle of rotation, which is just the intersection of the plane of rotation with the sphere. The key properties are

• Two circles of rotation
• The two circles of rotation are at right angles to one another
• Each point on each circle is at the same distance from the each point on the other circle.
• Every point on the 3-sphere revolves in a circle. Each such circle has the same radius

Imagine an ordinary cylinder in 3D, with the usual axis of rotation drawn as a line. There is a circle at each end where the surface takes its right angle bend. Identify each circle with one of the two 4D circles of rotation, and identify that 3D axis of rotation line with the point at the center of the 3-sphere. Now to get the planes at right angles, curve the cylinder 90 degrees. If we measure the distance between the circles in angular units, then we've satisfied all four criteria.

The model is geometrically meaningless and doesn't scale up to higher dimensions. Let's try for something a bit more geometric. In a 30-sphere the area near the circles of rotation is small and the maximum area is at a sort of "latitude" which is half way between them. We can make a model that is at least roughly accurate like this.

Imagine an ordinary cylinder in 3D. There is a circle at each end where the surface takes its right angle bend. Identify each circle with one of the two 4D circles of rotation.
Then shrink both of these circles to a point, thus transforming the cylinder into a 2-sphere. Note that every plane that passes through the origin of this sphere defines an orbit in 4D, except for the planes that pass through the circles of rotation which we forced to be points. Since these two points are each zero dimensional, no angle between them is defined, so we can say that they are at right angles to one another :-).

This model has the "advantage" that it looks like the familiar 2-sphere, with the difference that there are only 90 degrees of latitude instead of 180. We have a well-defined many-to-one mapping between the between the 3 and 2 spheres, because we can describe each point in each with three polar coordinates. Each orbit in the 2-sphere has the same tilt and a phase as the corresponding orbit in the 3-sphere. Details later. It isn't an isomorphism because the circles can intersect. We can fiddle with the model to get an isomorphism by giving each orbit a unique altitude above the surface of the 2-sphere. This altitude can be in a tiny range so that it looks identical. But there is another way. Instead we can animate the model so that while points can occupy the same space momentarily it will still be obvious which point is which by the way they move.

Now let'svisualize the two rotations with different periods. Each orbit can rotate in two different ways. Each orbit is a circle so it can revolve around its own center. Each orbit can also revolve around the axis of the sphere, which is called precession. The periods of those rotations can be anything, and those rotations can even be in opposite directions.

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Details:
"Each orbit in the 2-sphere has the same tilt and a phase as the corresponding orbit in the 3-sphere." Instead of R^4 let's use two unit discs in C^2. Each point is x and y. x = ze^it, y = (1-z^2)e^iu, with 0<=z<=1. For each circular orbit t is radians along the orbit, u is phase, and tilt is arcos(z).
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There are some attempts to visualize a 3-sphere, here

Here's a better way to visualize rotation of a 3-sphere.

The 3-sphere can be parameterized with polar coordinates as sin(x)e^it, cos(x)e^iu, with 0<=x<pi/2, -pi<=t,u<pi. Map to the three axis of R^3 using x, sin(x)t, and cos(x)u.

One may think of this as specifying a rectangle for each value of x, with the rectangles stacked along the x axis. For each such rectangle rotation causes the points to move in a diagonal direction. This direction is the same for every point in that rectangle. Each rectangle is topologically a torus, so movement over an edge means reentry over the opposite edge.

I can't imagine what this would look like.

1. What is a higher dimensional sphere?

A higher dimensional sphere is a generalization of the familiar 3-dimensional sphere to a higher number of dimensions. It is a set of points in a higher dimensional space that are equidistant from a central point, just like how points on a 3-dimensional sphere are equidistant from its center.

2. How can we visualize higher dimensional spheres?

Higher dimensional spheres cannot be visualized in the same way as 3-dimensional spheres. However, we can use mathematical representations and projections to visualize them in lower dimensions. For example, a 4-dimensional sphere can be represented as a 3-dimensional object by using a projection onto a 3-dimensional space.

3. What are some real-world applications of visualizing higher dimensional spheres?

Visualizing higher dimensional spheres can be useful in various fields such as physics, computer science, and mathematics. In physics, they can be used to study the properties of higher dimensional spaces and in computer science, they can be used in data analysis and machine learning algorithms. In mathematics, higher dimensional spheres have important applications in algebraic topology and geometry.

4. Can we visualize higher dimensional spheres in our everyday life?

No, we cannot directly visualize higher dimensional spheres in our everyday life as our perception is limited to 3 dimensions. However, we can use analogies and mathematical representations to understand their properties and visualize them in lower dimensions.

5. Are there any limitations to visualizing higher dimensional spheres?

Yes, there are limitations to visualizing higher dimensional spheres. As the number of dimensions increases, it becomes more difficult to visualize and understand their properties. In fact, our brains are not wired to imagine spaces with more than 3 dimensions, making it challenging to visualize higher dimensional spheres accurately.

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