How was the 3D model of the Tesseract and other 4 dimensional shapes made?
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Anything that is drawn on paper is effectively two dimensional. Despite that, it’s easy to draw a cube on paper in such a way that your brain immediately makes the leap the thinking about it as a three dimensional object. This is due to two things:
– The drawn “cube” is not a faithful reproduction of a cube (it can’t be), but it is faithful in all the right ways. It is a limited visualization that nevertheless conveys characteristics of a 3 dimensional object in just two dimensions.
– Your brain has phenomenal 3d intuition, and can easily *interpret* a such a drawing as a three dimensional object.
The first point is exactly analogous to 3D models of 4D objects. The second point very much isn’t.
If you draw a cube on paper, the points sharing an edge aren’t all the same distance from each other. That’s a property of the cube that *isn’t* faithfully captured by our drawing. Our brains have the 3d intuition to immediately realize that the given drawing *represents* an object where all points that share an edge *are* the same distance from one another.
We know how 4 dimensional objects, such as the 4-sphere or the Klein bottle, behave. In a sense, we know how they “look”. Many of these objects are actually very easy to fully describe mathematically.
The problem is that our brains refuse to visualize them. Nevertheless, we can convey certain aspects of their “appearance” in 3D space, precisely the way we can draw a cube on paper. These attempts are imperfect, however, as in 4 dimensions our brains do not have the intuition to reconstruct the properties that our limited representation doesn’t faithfully capture.
Anything that is drawn on paper is effectively two dimensional. Despite that, it’s easy to draw a cube on paper in such a way that your brain immediately makes the leap the thinking about it as a three dimensional object. This is due to two things:
– The drawn “cube” is not a faithful reproduction of a cube (it can’t be), but it is faithful in all the right ways. It is a limited visualization that nevertheless conveys characteristics of a 3 dimensional object in just two dimensions.
– Your brain has phenomenal 3d intuition, and can easily *interpret* a such a drawing as a three dimensional object.
The first point is exactly analogous to 3D models of 4D objects. The second point very much isn’t.
If you draw a cube on paper, the points sharing an edge aren’t all the same distance from each other. That’s a property of the cube that *isn’t* faithfully captured by our drawing. Our brains have the 3d intuition to immediately realize that the given drawing *represents* an object where all points that share an edge *are* the same distance from one another.
We know how 4 dimensional objects, such as the 4-sphere or the Klein bottle, behave. In a sense, we know how they “look”. Many of these objects are actually very easy to fully describe mathematically.
The problem is that our brains refuse to visualize them. Nevertheless, we can convey certain aspects of their “appearance” in 3D space, precisely the way we can draw a cube on paper. These attempts are imperfect, however, as in 4 dimensions our brains do not have the intuition to reconstruct the properties that our limited representation doesn’t faithfully capture.
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