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Fondamenti Di Elettronica By Muhammad Rashid Ita Pdf urbanio 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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