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How should you cut a cube with a single planar cut so that one face is

  • a regular hexagon?
  • a regular octagon?

How should you cut a torus with a single planar cut so that you get

  • two same-radius circles with different centres?
  • three circles?
  • ovals, like the circles?
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  • $\begingroup$ You deleted your previous puzzle? :-( $\endgroup$
    – Rand al'Thor
    Sep 25, 2016 at 1:16
  • 2
    $\begingroup$ @randal'thor Yes, it contained no trace of soul uplifting beauty, but very ugly and clunky. $\endgroup$
    – user29273
    Sep 25, 2016 at 1:17
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    $\begingroup$ @tpk Awesome reason to delete a question. I wonder what happened of everyone here did that. $\endgroup$
    – user27395
    Sep 25, 2016 at 5:07
  • $\begingroup$ I agree, but I don't think that puzzle actually merited deletion. $\endgroup$
    – Gareth McCaughan ♦
    Sep 25, 2016 at 15:08

2 Answers 2

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Cube

Hexagon:

cube

Octagon:

impossible, because a cube only has 6 faces and so we can only form polygons with at most 6 sides by taking single planar cuts through a cube.

Torus

Concentric circles:

torus

Non-equal same-radius circles:

torus

(I already knew how to do this, but for ease of answering I sourced the images from here and here.)

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  • $\begingroup$ Oh I miswrrote the question. See it again. $\endgroup$
    – user29273
    Sep 25, 2016 at 1:06
  • $\begingroup$ What bout the Other ones ? $\endgroup$
    – user29273
    Sep 25, 2016 at 1:12
  • $\begingroup$ @tpk Are you sure an octagon and three circles are possible? $\endgroup$
    – Rand al'Thor
    Sep 25, 2016 at 1:14
  • $\begingroup$ Hint - why the mathematics tag ? Visual tag would suffice. $\endgroup$
    – user29273
    Sep 25, 2016 at 1:25
  • $\begingroup$ @humn Oops. What I really meant was non-equal. I really should be in bed by now. $\endgroup$
    – Rand al'Thor
    Sep 25, 2016 at 1:27
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Two circles

A plane through the centre of the torus, perpendicular to the hole.

Two ellipses

Same, but not through the centre.

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