# Cutting a cube and a torus

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?
• You deleted your previous puzzle? :-( Commented Sep 25, 2016 at 1:16
• @randal'thor Yes, it contained no trace of soul uplifting beauty, but very ugly and clunky.
– user29273
Commented Sep 25, 2016 at 1:17
• @tpk Awesome reason to delete a question. I wonder what happened of everyone here did that.
– user27395
Commented Sep 25, 2016 at 5:07
• I agree, but I don't think that puzzle actually merited deletion. Commented Sep 25, 2016 at 15:08

## Cube

Hexagon:

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:

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

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

### Two circles

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

### Two ellipses

Same, but not through the centre.