r/Geometry

In this video I go through the common 4D shapes, the tesseract and the pentatope, and show how they can perfectly represent what I propose as a better regard of the fourth dimension, which is the spatial capacity for motion.

Video Transcript:

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The tesseract is the 4D progression of a cube. Just as a cube is built from two squares with new lines connecting them in a new direction, the 4D tesseract is built from two cubes connected in what has been speculated as a new fourth direction orthogonal to 3D space.

What I’ve considered is not a new purely spatial direction that geometers still teach about, but instead a spacetime axis representing motion. Herman Minkowski and Albert Einstein popularized time as a fourth dimension, which paired with space gives us spacetime, which enables the spatial capacity for motion. Here the tesseract can perfectly represent a cube expanding and contracting.

This is another depiction of a tesseract, or hypercube, described as two cubes facing each other with all vertices connected by new 4D edges.

Instead of the expansion and contraction motion of the last example. This hypercube can represent simple positional motion from one space and time to another.

The pentatope is the dimensional progression from a triangle to a tetrahedron. The "4D edges" are added as lines from each vertex toward the center.

While the tesseract continues a cube's pattern of symmetry, expanding all part equally, the pentatope continues a tetrahedron's pattern of simplicity, moving only one vertex.

For more than 150 years since Hinton, the fourth spatial dimension has been presented in science education and popular media as an abstract, unreachable direction at a new right angle to length, width, and height — something mathematically required but physically impossible to visualize.

My proposal is that this long-standing teaching is not necessary.

The logic and math are sound, but there's a conceptual and philosophical gap that has never been properly addressed. Instead of chasing an abstract fourth spatial direction, we can understand 4D geometry far more simply by blending it with Minkowski's spacetime, showing it naturally as the geometric record of ordinary three-dimensional objects moving through space and time.

Minkowski came up with the worldline, which already does this — the 4D history of an object through spacetime — when he and Einstein popularized time as the fourth dimension and paired it with 3D space to describe 4D spacetime.

Developing my own idea, I've shown how the familiar tesseract — a cube within a cube connected by "4D edges" — can be seen as the worldline, or 3D spacetime-map, of a single cube undergoing motions of expansion or contraction. The so-called "new edges" are not a hidden fourth direction; they are displacement vectors tracing the paths of motion — along what we should perhaps more accurately term a spacetime axis as the true fourth spatial axis.

This idea works with other 4D forms like the pentatope. The tesseract continues a pattern of symmetry from lower dimensions — a square with equal sides and angles, a cube with equal sides and angles, and now a uniformly expanding or contracting cube. The pentatope (4-simplex) offers a complementary case continuing a different pattern — minimalism rather than symmetry. Where the tesseract represents uniform motion of all parts of a cube, the pentatope can represent the motion of just a single vertex of a tetrahedron moving inward toward its center. That path of motion shapes exactly the 3D projection of the pentatope — a tetrahedron with "4D edges" from each vertex meeting at the center. In this example there is just one displacement vector, as the other "4D edges" are simply part of the new shape.

So to put it plainly, 4D polytopes, and even 4D forms such as the 3-sphere, are simply better understood as worldlines — representing a 3D shape's transformation through space and time, with the "4D edges" as displacement vectors showing paths of motion. This representation is more intuitive, more physically meaningful, and more aligned with how we already use 4D coordinate systems in practice — in animation, physics engines, robotics, VR, and holography. As these fields grow, I wonder whether we may eventually come to treat the motion of 3D forms as the true and intuitive meaning of 4D geometry, rather than continuing to picture the fourth dimension as an unreachable static direction.

It is a bold opinion, and I think the unreachable 4D static spatial axis should be done away with. Let's stop chasing or trying to explain it. 4D geometry accomplishes something far more useful by representing the motion of 3D forms through spacetime with precision and elegance. I recognize that Euclidean 4D space and Minkowski spacetime have different metrics — this is a pedagogical reinterpretation, not a claim that they are mathematically identical. 

I am not an academic, but I have been thinking about this over the past few years and would appreciate some sincere feedback. I've tried presenting it to academic journals, but I lack the mathematical formalism that most prefer. I'm aware this is more philosophical and geometric interpretation than rigorous new mathematics. Does this interpretation seem meaningful, misguided, redundant, or potentially formalizable?

I am also surprised at how little discussion there is on this, mainly from the pure geometry crowd. Physicists happily work with similar conceptions through 4D spacetime, and 3D graphics programmers regularly treat motion through 4D geometric systems, but geometers still teach the static axis of Hinton's tesseract. Is it time to move on? I think so, and that this reinterpretation is very long overdue.

Hello r/geometry! I took a picture of a rectangle that I know to be exactly twice as wide as it is tall. I took the picture at a very much not-head-on angle, and I would like to know the exact angle I took it at in three dimensions, relative to the rectangle itself. For example, if I had to guess, I was ~10 degrees above head-on, ~70 degrees to the right, and rotated ~5 degrees clockwise from the rectangle, but I'd like to be able to calculate those angles more confidently by taking apparent angle and side length measurements from the picture of the rectangle.

Is this possible? Is there a formula that I would be able to understand well enough to calculate my perspective from a bunch of pictures from different angles? I'm asking this as someone with a respectable math education mostly in statistics.

Suppose we want to obtain a figure from the sum of three numbers such that a^n+b^n=c^n. If we consider n dimensions starting from n=1, we obtain:

n=1: (It's impossible to obtain a figure even if, for example, a^1+b^1=c^1.)

n=2: Possible! (Right-angled triangle).

n=3 (Impossible?) The figure would be a triangular prism, but it turns out that a^3+b^3<>c^3. That is, the sum of the volumes of the cubes built on the legs is not equal to the volume of the cube built on the hypotenuse!

n=4 ?

Why does it seem to be possible only for n=2?