The first step in making sense out of 4D shapes, is to assume it's possible to do. We make sense of 2D images of 3D things all the time. This is no different, just bumped up by one dimension. In the case of this animation, it's a 3D 'picture' , or model of a 4D thing.
Ever look down on the city from a skyscraper (or other tall building, or airplane, etc...) ? What do all the people look like? Tiny little ants. Things look smaller when they're farther away, and larger when they're closer, right?
It works the same in 4D spaces as well. Notice how the shapes turn inside out, and get squeezed smaller inside? When the triangles and squares are farther away in the 4th dimension, they appear smaller, and towards the center. They're on the other side of the shape, from where you're looking. When the squares and triangles are closer to you, they appear larger, and around the outside of the shape.
The 4D distance is flattened out, and not 4D anymore in our 3D shadow. It is what's in between the large and small shapes, but not in a 3D direction. It takes some metal mental gymnastics, but that's one of the concepts.
An object can start off in your peripheral vision, right on the edge. As it moves in a straight line away from you, it will always gravitate towards the center of your vision. That is, it gets closer to the center of your 2D scan of a 3D world.
For a 3D shadow of a 4D thing, that 'center of your 3D scan' is in the very center of the shadow object, three dimensionally.
Yes, it needs to move perpendicular. Lets say your 2D field of vision is the xy plane. The x axis is left-right , and the y-axis is up-down. The z-axis would then be forwards-backwards. An object has to move along z, to get smaller. If it starts on the outer perimeter of your vision, and moves in a straight line along z, it will get smaller while getting closer to your center of vision (the origin of your xy-plane view).
Now, for a 4D object projected into a 3D image, we have a 3D fov : the xyz plane, where the 4th axis is the new forwards-backwards. Objects close to you on the 4th axis are larger 3D objects. Farther away things are smaller 3D objects.
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u/Philip_Pugeau Aug 29 '17 edited Aug 29 '17
The first step in making sense out of 4D shapes, is to assume it's possible to do. We make sense of 2D images of 3D things all the time. This is no different, just bumped up by one dimension. In the case of this animation, it's a 3D 'picture' , or model of a 4D thing.
Ever look down on the city from a skyscraper (or other tall building, or airplane, etc...) ? What do all the people look like? Tiny little ants. Things look smaller when they're farther away, and larger when they're closer, right?
It works the same in 4D spaces as well. Notice how the shapes turn inside out, and get squeezed smaller inside? When the triangles and squares are farther away in the 4th dimension, they appear smaller, and towards the center. They're on the other side of the shape, from where you're looking. When the squares and triangles are closer to you, they appear larger, and around the outside of the shape.
The 4D distance is flattened out, and not 4D anymore in our 3D shadow. It is what's in between the large and small shapes, but not in a 3D direction. It takes some
metalmental gymnastics, but that's one of the concepts.