Shortest Distance On Tesseract

Geometry Level 5

Brilli the Ant is trapped on the \(2D\) surface of the villian's tesseract, which has an edge length of \(42\text{ mm}\). He is at \(4 \text{ mm}\) from the center of one of the square faces, as measured parallel to the sides edges. He needs to find the shortest path on the \(2D\) surfaces of the tesseract to the center of the face that is opposite of the face he is on now, where there is a micro-wormhole he can use to escape from the tesseract.

Find this shortest distance in \(\text{ mm}\).

Note: We could say the vertices of the tesseract are at \((0,0,0,0), (0,0,0,42), (0,0,42,0), ....,(42,42,42,42)\), so that Brilli the Ant would be at \((21,17,0,0)\), and the micro-wormhole he needs to get to would be at \((21,21,42,42)\). He can travel only on the surfaces of the tesseract, i.e. at all points of its path, \(2\) of the point coordinates must either be \(0\) or \(42\).

The surfaces of the tesseract appear kind of like flat soap bubbles in this animation, bounded by straight edges.


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