what the 3d representation of a tesseract is actually showing
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It shows every side of the cube. Its a weird concept to wrap your head around but if you were to rotate it it would apear to warp through itself. Basically similar if you would be a 2d object you could never see all sides of an object, some would be concealed.
But as a 3d object you can look from the top of the 2D object and see all the sides, none are hidden.
So tessaract would be like a 4d object or person looking at a cube and seeing all 6 sides
The shadow of a 4D cube analog as cast onto a 3D surface.
It behaves in the same way as if you were to make a wire-frame cube and look at its 2D shadow. You know that the cubes sides are all squares. But the shadow does not show squares, it shows quadrilaterals with lengths and angles that change as you rotate the wire frame, as well as them going through eachother.
You have enough experience with cubes and shadows though, that you are able to infer the shape of the 3D object from its shadow. You are not able to do the same with a 4D object.
The 3D render shows you warped cubes (the “sides” of the 4D object), changing in size and angles, as the 4D object is rotated. These also go “through” eachother in the render.
Here’s a really good blog post that demonstrates how it works: https://ciechanow.ski/tesseract/
If you take a 3D cube and you project it onto a plane (like by casting a shadow), you’ll get a 2D image. The 3D representation is the same idea, but projecing a 4D object into a 3D space.
A 3d representation of a tesseract shows you information about the tesseract. We can’t see a tesseract, the only thing we can see is a 3D object (a cube). So you need to go through a mental exercise to visualize a tesseract.
The mental exercise is as follows:
* imagine that you’re restricted to only seeing 2D images, shadows on a paper.
* realize that rotating an object, especially rotating in around the extra axes that you don’t have available on the 2D paper shadow, deforms the shadow in ways that give you a lot of information about the object.
So you KNOW that the object must have squares as its sides, but from looking at the paper you can’t *see* the *vertical* way in which those squares “combine” into a cube. But rotating the cube shows you a lot of information about the wireframe of the cube, in the way your square shadows deform into trapezoids and other such shapes, rather than remaining squares.
As far as seeing a tesseract, get a cubic meter cardboard box and look at it for 3.33 nanoseconds, and what you will see is a tesseract. The xyz sides of the tesseract are 1 meter, and the 4th dimension is ct = 1 meter.
The 3D representation of a tesseract is showing the extension of the cube into a fourth spatial dimension. You can see it has height (up & down dimension), width (side to side dimension) and depth (back and forth dimension). But, it also has another measurement that I’m going to call “wumbo-ness” that extends ana and kata into the 4th dimension.
Just like a 2D drawing of a 3D cube has lines from the corners of the 2D square projected diagonally to represent extension into the third dimension, a 3D representation of a 4D cube has lines that come off the corners at diagonals to represent extension into the fourth dimension.
The 2D drawing looks like the shadow of the 3D object. The 3D representation of the tesseract would be the 3D “shadow” of the 4D object.
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Adding on, even though we can’t perceive the 4th dimension with our eyes shouldn’t we still be able to feel 4D objects? With our hands for example
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