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Can someone please help explain what the coordinates of a cube of length one at a quarter of it's body diagonal would be? I know that the coordinates of the center of a unit cube are (1/2, 1/2, 1/2) but what would be the coordinates at a quarter of it's body diagonal, I think it could be
(1/4,1/4,1/4) but I have no logic to prove this, can someone please help?

Thanks in advance!

I've only added one tag for this question, feel free to add any as applicable, thank you.

harry
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    It comes down to two questions: 1) is the point $(\frac14,\frac14,\frac14)$ lies on the diagonal? 2) is the distance between $(\frac14,\frac14,\frac14)$ and an endpoint of the diagonal one fourth the length of the diagonal? If both answer is yes, then you have verified $(\frac14,\frac14,\frac14)$ is at the quarter of a space diagonal (with respect to above endpoint). – achille hui Oct 28 '21 at 00:16
  • @achillehui, I assumed since the length of the cube is 1, two coordinates diagonally opposite could be (0,0,0)->(1,1,1) hence ( 1/4,1/4,1/4) would lie on the diagonal? But the distance between them is not one fourth the length, using your logic I get (3/4,3/4,3/4), can this be correct? – harry Oct 28 '21 at 00:25
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    there are two end points of the diagonal: $(0,0,0)$ and $(1,1,1)$. The answer depends on which endpoint you consider to be the start. if $(0,0,0)$ is the starting end point, then $(\frac14,\frac14,\frac14)$ is the answer. otherwise, the answer is $(\frac34,\frac34,\frac34)$. – achille hui Oct 28 '21 at 00:43
  • @ harry: go on to next, half of half is quarter of diagonal. – Narasimham Oct 28 '21 at 00:51
  • @achillehui, since I'm not given a starting point just a length it would make sense to make (0,0,0) my staring point? Which would mean it makes sense for (1/4,1/4,1/4) to be the point that lies on the quarter of the body diagonal, thank you! – harry Oct 28 '21 at 01:01
  • @Narasimham, are you saying it should be (1/4,1/4,/14)? – harry Oct 28 '21 at 01:01
  • You kept the origin fixed, right? Now can you found out the simlar triangle half sized? – Narasimham Oct 28 '21 at 01:09

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