What is a positive 600 cell body? The positive 600 grid is actually a four-dimensional convex positive grid consisting of 600 regular icosahedral bodies, 1, 200 regular triangular faces, 720 sides and 1, 20 points. Its Shreveley symbol is {3,3,5}, which is the duality of normal 120 lattice, that is, the opposition between row vector and column vector.
For people who live in three-dimensional space, four-dimensional space is unimaginable, which is similar to Klein bottle, but just as we can imagine three-dimensional objects by studying three-dimensional projections on two-dimensional planes, we want to imagine four-dimensional space, and we can also study it by using four-dimensional projections in three-dimensional space.
Who invented the positive 600 cell body? Regular 600 cell is a kind of regular cell in topology, which is classified as a four-dimensional cube in topological space, and topological space was first proposed by French mathematician Frey in 1906, so in a sense, regular 600 cell originated from Frey, but the specific inventor of regular cell is still unknown.
Projection of regular 600 cells 1. Schlegel projection: regular 600 cell is a four-dimensional geometric solid figure, so it can be studied by schlegel projection, but compared with the simplicity of three-dimensional projection, four-dimensional projection is much more complicated. For example, the projection of the conventional 600 cell is actually projected right against the vertex.
2. Spherical projection: Of course, in addition to four-dimensional projection into three-dimensional, there is another one that is obtained by expanding each face of a hypercube? Super ball? Spherical projection refers to the parallel projection we see in the outermost part.
3. Two-dimensional wireframe orthographic projection: that is, the projection of positive 600 cells seen on the two-dimensional plane. Because it is two-dimensional, only some lines and points will appear. The red dot in the figure refers to the part where two points overlap, and the yellow dot refers to the part where four points overlap.
Contact with the positive 120 unit: make the center of each regular tetrahedron in the positive 600 unit perpendicular to the regular triangular surface of the regular tetrahedron, and you can get the positive 120 unit, while the positive 120 unit can get the positive 600 unit through similar processing. That is to say, taking the center of each regular dodecahedron in the regular 120 cell body as the vertical line of the regular dodecahedron regular Pentagon where the center is located, we can get the regular 600 cell body.
In the negative dimensional space, it is mentioned that the positive cell is a four-dimensional figure, so the positive 600 cell is actually the product of four-dimensional space, so it is difficult for us to understand it with the thinking of three-dimensional space, and we can only better understand it with the projection method in mathematics.