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pdfWernher von Brauns Rocketry-Altruismus-1961-Deutsch-Erweiterte-Urtext-version.pdfQ: Counting number of cubes in 3D space I was wondering if there is any easy way of counting how many cubes are there in 3D space? Consider the cube $[0,1] imes[0,1] imes[0,1]$. This represents the unit cube, the 3D space. It is possible to transform a cube into a different one by rotating and scaling. However, I am not really interested in rotations. For example, given two cubes $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$, does there exist an affine transformation such that the two cubes are identical? By "affine transformation", I mean a function that takes a point (a point $(x,y,z)$ is a point with an x, y, z coordinate) in the 3D space and outputs a point in the same space. A linear transformation is a 1D affine transformation, a 2D affine transformation is a linear transformation followed by a 1D affine transformation, and a 3D affine transformation is a 2D affine transformation followed by a 1D affine transformation. A: Let $S^n$ be the $n$-dimensional sphere and $S^n_q$ be its equator of height $q$, that is, the sphere of radius $q$. Then you have the following characterization. The number of cubes of side-length $q$ in $S^3$ is the number of cubes of side-length $q$ in $S^3_q$. To see this, take a cube of side-length $q$ in $S^3_q$ and note that it is also a cube of side-length $q$ in $S^3$. Now let's prove the claim by induction on $q$. Let $Q$ be a cube of side-length $q$ in $S^3$. Its image in $S^3_q$ is a cube of side-length $q$ in $S^3_q$ and its image in $S^3_{q-1}$ is a copy of $Q$ on the equator

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