Slicing donuts and sugar cubes

[bonanova]

If a plane slices a polyhedron, the cross section is a polygon. Think of slicing a brick of butter or cheese.

If a plane slices a cone or cylinder, the cross section is an ellipse.

Let's think about ways to create regular polygonal [e.g. square] and regular elliptical [circular] cross sections:

.

1. How many different regular polygons can be made by slicing a cube?
2. How many different ways can you slice a torus [think doughnut] to obtain circles?
   .
   

Are the answers the same?

[superprismatic]

1. Infinite. You can make one square an infinite number of ways -- the size of a face; and you can make an infinite number of different sized triangles.

2. Infinite. You can make two disjoint circles an infinite number of ways and an infinite number of different sized concentric circles.

Yes, they are both of size Aleph-one.

Edited November 4, 2010 by superprismatic

[bonanova]

Once you make a square, another square is not a different regular polygon.

Consider fundamentally different [ways of making] slices.

[superprismatic]

Once you make a square, another square is not a different regular polygon.

Consider fundamentally different [ways of making] slices.

By this logic, the answer to part 2 must be 1 because once you make a circle, another circle is not a different regular elliptical.

Is that correct?

[bonanova]

By this logic, the answer to part 2 must be 1 because once you make a circle, another circle is not a different regular elliptical.

Is that correct?

Well, you're correct in observing there is only one regular ellipse.

But the answer to part 2 is not 1.

[1 ] Different regular polygons, and [2] different ways of making circles are requested.

Regarding [2], fundamentally different is meant to say that slicing a doughnut with a plane containing the axis is to be counted as [just] one way. You can rotate the plane about the axis through an infinity of angles, but the result will be fundamentally the same. If it helps in describing the results, you can assume the doughnut is an ideal torus with inside diameter a and outside diameter 3a.

[superprismatic]

Well, you're correct in observing there is only one regular ellipse.

But the answer to part 2 is not 1.

[1 ] Different regular polygons, and [2] different ways of making circles are requested.

Regarding [2], fundamentally different is meant to say that slicing a doughnut with a plane containing the axis is to be counted as [just] one way. You can rotate the plane about the axis through an infinity of angles, but the result will be fundamentally the same. If it helps in describing the results, you can assume the doughnut is an ideal torus with inside diameter a and outside diameter 3a.

1. 2 -- triangle and square (ignoring the fact that a square can be made in 2 fundamentally different ways: with a plane parallel to a face and with a plane at a 45° angle to a face)

2. 2 -- one making two concentric circles, and one making two separated circles.

Yes, the answers are the same.

[plainglazed]

triangle, quadrilateral (square, rectangle, parallelogram), hexagon, and I think maybe a pentagon but cant quite visualize it fully or draw it well enough

[bonanova]

Progress, but no correct answers yet. Well, one correct answer.

[araver]

1. Answer: 3 - Equilateral triangle, square, regular hexagon.

Regular pentagon isn't possible (God, I sure wanted to slice a cube into a regular pentagon when I was a kid , my dad was unfortunately right) and 7 or more edges means more than 6 faces which the cube hasn't.

2. Answer is a bit trickier as it makes a small assumption on the definition of "slicing".

Put doughnut on table:

Way 1 - slice it like a knife slicing bread in a perpendicular motion, making sure you start over/intersect the empty middle of the torus, and you get 2 circles (equal).

Way 2 - slice it parallel to the table to get 2 circles (concentric)

Way 3 - "slice" it parallel to the table just on the top to get 1 circle. Actually that's not what I'd normally call slicing, it's more like putting a sheet of paper on top or see the leaking circle of a doughnut on the paper, but I think it should count as a "slice" as there is nothing to contradict this in the OP. (If the wording were "plane intersecting a torus", way 3 is valid)

Can't think of other ways to get circles, so I'd say 3 different ways.

3=3

q.e.d.

[bonanova]

1. Answer: 3 - Equilateral triangle, square, regular hexagon.

Regular pentagon isn't possible (God, I sure wanted to slice a cube into a regular pentagon when I was a kid , my dad was unfortunately right) and 7 or more edges means more than 6 faces which the cube hasn't.

2. Answer is a bit trickier as it makes a small assumption on the definition of "slicing".

Put doughnut on table:

Way 1 - slice it like a knife slicing bread in a perpendicular motion, making sure you start over/intersect the empty middle of the torus, and you get 2 circles (equal).

Way 2 - slice it parallel to the table to get 2 circles (concentric)

Way 3 - "slice" it parallel to the table just on the top to get 1 circle. Actually that's not what I'd normally call slicing, it's more like putting a sheet of paper on top or see the leaking circle of a doughnut on the paper, but I think it should count as a "slice" as there is nothing to contradict this in the OP. (If the wording were "plane intersecting a torus", way 3 is valid)

Can't think of other ways to get circles, so I'd say 3 different ways.

3=3

q.e.d.

Nice! ... but ...

I'll share with you that often the hardest part of framing a puzzle is to cover all the contingencies without using a hundred sentences. The great puzzle maker Martin Gardner did it best; it was part of his genius, apart from the puzzles themselves. You touched a base that I didn't cover. Nice going. Now I'll cover it.

I hadn't thought of getting just one circle. It may be a philosophical issue whether a slice can ever become a line. But for the purposes of this puzzle, I'll define a slice as an operation that leaves portions of the object being sliced on both sides of the slicing plane. I don't mean to change the puzzle midstream, I just envisioned, but did not clarify, that a slice would divide the object into two parts, creating opposing faces, which if glued and then joined, would restore the object. I think that rules out getting just one circle; you'd always get two. Not two in the sense of the two opposing faces, two in the sense that you described.

And, if we're tracking progress, had one correct answer; now there are two. Huh?

Yep, two, and that's a clue.

As well as a rhyme.

[plainglazed]

thought I had the pentagon visualized but read the other posts and realize had overlooked that "regular" bit. thinking then superprismatic's two is right for 1. but only see two pair of circles for the torus. still pondering

[araver]

I think that rules out getting just one circle; you'd always get two.

Then with this clarification in mind, my answers are 3 to the first question (equilateral triangle, square, regular hexagon) and 2 to the second question (pair of concentric circles, pair of disjoint equal circles).

And I'll add for answer 1 the reason why you cannot get a regular pentagon (forgot to say in my post) by slicing a cube. Slicing creates edges on different faces of the cube. Obtaining a pentagon means 5 edges on 5 different faces of the cube. Since there are 3 pairs of opposite parallel faces of the cube, from Dirichlet's principle out of 4 edges two are on opposite faces of the cube. These opposite edges are in parallel planes and belong to the same plane (slicing plane) hence they are parallel themselves. Regular pentagons have no pair of parallel edges, q.e.d.

Edited November 5, 2010 by araver

[bonanova]

We're back to one correct answer [of polygons / circles / equal].

[Guest]

We're back to one correct answer [of polygons / circles / equal].

OK, just so we're clear:

1. We're talking about the intersection of the given shape and a single plane. So, you only get one chop.

2. We're talking about a "regular" torus here. As in, one where the axis of revolution does not intersect the cross-section?

[bonanova]

OK, just so we're clear:

1. We're talking about the intersection of the given shape and a single plane. So, you only get one chop.

2. We're talking about a "regular" torus here. As in, one where the axis of revolution does not intersect the cross-section?

Yes. And yes.

Sorry to make a deceptively simple puzzle sound unclear.

[Guest]

Yes. And yes.

Hmmm... I thought so, but figured I'd make sure. From what I can deduce, the correct answers thus far are "3" and "yes", but I just don't see it for the torus... :headscratch:

[bonanova]

Hmmm... I thought so, but figured I'd make sure. From what I can deduce, the correct answers thus far are "3" and "yes", but I just don't see it for the torus... :headscratch:

Assuming you're correct, and that's a pretty safe assumption, one can ask the orientation of the third slicing plane. Lay the doughnut on a table and think. Symmetry probably demands that any slice that makes circles must contain the center of the torus. There is only one type of vertical cut, that makes two separated circles. There is only one type of horizontal cut, that makes concentric circles. That exhausts the orthogonal cuts. What's left?

And now begins the creative visualization. The answer is both unexpected and beautiful.

[bonanova]

1. 2 -- triangle and square (ignoring the fact that a square can be made in 2 fundamentally different ways: with a plane parallel to a face and with a plane at a 45° angle to a face)

2. 2 -- one making two concentric circles, and one making two separated circles.

Yes, the answers are the same.

I've been thinking about your answer.

Yes there are two [fundamentally] different ways to make a square. I hadn't seen that.

The wording of the OP lumps them together, as you rightly conclude.

We're looking for ways that a slice makes different polygons.

Sorry not to reply sooner.

[araver]

Assuming you're correct, and that's a pretty safe assumption, one can ask the orientation of the third slicing plane. Lay the doughnut on a table and think. Symmetry probably demands that any slice that makes circles must contain the center of the torus. There is only one type of vertical cut, that makes two separated circles. There is only one type of horizontal cut, that makes concentric circles. That exhausts the orthogonal cuts. What's left?

And now begins the creative visualization. The answer is both unexpected and beautiful.

there is a 3rd way to slice through the center of the torus giving 2 intersecting circles

Which makes both answers equal to 3.

Unfortunately, I'm not very good with 3D graphics so I can only show the cut in a 2D slice, going through the center of the torus and tangent to the torus both up and down.

If the outside (Do) and inside(Di) diameters are 3a and a then the angle is 30 degrees as the picture below.

Otherwise, the angle is arcsin((Do-Di)/4/[(Do-Di)/4+Di/2]).

Edited November 6, 2010 by araver

[Guest]

Correct me if I'm wrong, but doesn't your solution only create one circle? Since both your circles share two common points, it is logically impossible for them to be two unique circles; one of them must be a false circle.

[araver]

Correct me if I'm wrong, but doesn't your solution only create one circle? Since both your circles share two common points, it is logically impossible for them to be two unique circles; one of them must be a false circle.

Not really. I did not say the circles are tangent to one another.

What I said is that the 2 circles have exactly 2 points in common (like 2 adjacent Olympic circles):

- both points are on the interior of the torus

- one is on the upper half of the torus

- the other is on the lower half of the torus, symmetrical with respect to the center of the torus.

The circles are equal to one another.

The centers of these circles don't seem to have a very special locus, but maybe it helps to imagine 4 more points on these circles.

If you put a stick through the center of the doughnut, in a plane parallel to the table, perpendicular to the cut I've shown, you get 4 more points. A,B,C,D (in order). A and D lie on the outside of the torus, B and C on the inside. A and C are on the same circle, B and D are on the second circle. Does this help?

Edited November 7, 2010 by araver

[Guest]

Ah, I see it now. But that leaves the question, and I'm no math whiz to figure it out, are those actually circles and not ovals?

[araver]

Ah, I see it now. But that leaves the question, and I'm no math whiz to figure it out, are those actually circles and not ovals?

they're circles alright.

The 4 points I described in the earlier post are on the same line with the two centers of the circles ... however to convince anyone I guess I'll have to figure a way to prove by equations (my experience with geometric equations is rusty, I'm afraid).

Edited November 8, 2010 by araver

[araver]

they're circles alright.

The 4 points I described in the earlier post are on the same line with the two centers of the circles ... however to convince anyone I guess I'll have to figure a way to prove by equations (my experience with geometric equations is rusty, I'm afraid).

for the case bonanova described (outer diameter is three times the inside diameter). In this case, the centers of the 2 circles are on the inside surface of the torus as well (points B and C in my previous posts).

Could not convince GeoGebra3D to make me a nice picture (although that environment led me to the existence in the first place).

Made a couple of pics using Moray.

[Guest]

Neato!

[bonanova]

Gorgeous pics, araver. Bravo!

They have names, actually, following the guy who first publicized them.

Wikipedia has a nice animation here.

Along with the math that proves they're circles, altho it could be hazardous to your health.

Aside: some other relevant and pretty pics.