# 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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