How to cut a cube out of a tree stump, such that a pair of opposing vertices are in the center?

I saw this picture of a cube cut out of a tree stump.

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I’ve been trying to craft the same thing out of a tree stump, but I found it hard to figure out how to do it.

One of the opposing vertices pair is on the center of the tree stump:

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I’ve been struggling to find the numbers and angle needed to make the cuts.

Any kind of advice would be greatly appreciated. Thanks for taking your time to read this and I’m sorry for my bad english.

Answer

First, form your stump into a cylinder with a height greater than $\frac{3}{\sqrt{2}}\approx 2.121$ times its radius so that the cube will fit inside the cylinder. Here, we assume you can mark given points and lines on the surface of the cylinder as well as cut a plane through $3$ points.

For the first vertex, mark the center of one of the cylinder’s circular faces. Next, from this point mark $3$ radii of the circular face spaced $120$ degrees from each other (so that their endpoints create an equilateral triangle).

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Now, go down from each of the $3$ vertices of the triangle a distance equal to $\frac{1}{\sqrt{2}}\approx 0.707$ the radius of the cylinder, and mark the points there. If your cylinder has the minimal height-to-radius ratio, these points will be exactly $\frac{1}{3}$ of the way down.

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Next, cut $3$ planes through the first vertex and each pair of the next $3$ vertices, forming the first corner of the cube.

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Now, mark the $3$ midpoints of the arcs formed by your cuts. These are the next vertices of the cube. Once again, if your cylinder has the minimal height-to-radius ratio, these will be $\frac{2}{3}$ of the way down.

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Finally, cut $3$ more planes to finish the shape of your cube. You do not need to mark the last vertex as it is predefined by the intersection of the planes.

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When you are all done, the side length of the cube will be $\sqrt{\frac{3}{2}}\approx 1.225$ times the radius of the starting cylinder, so take this into account if you want a specific side length. The angle formed by each face of the cube to the original end of the cylinder is $\arctan(\sqrt{2}) \approx 54.734^{\circ}$.

Attribution
Source : Link , Question Author : Aaron Cheuk , Answer Author : Joshua Wang

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