[math-fun] Sphere to cube dissection?
It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside. Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger. Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces?
... I didn't see the solution for 2D immediately ... but suddenly I did. Neat puzzle. On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside.
Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger.
Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I was an idiot not to look at the the 2D case first. I don't have a detailed solution, but the sketch is, a roughly square section in the middle cut into planks, which get reassembled end-to-end to make the outside of the square, and then the four remaining sections of the circle get chopped up fine enough to fill in the middle. I would guess it can be done in about 20 pieces at most. On Fri, Jan 12, 2018 at 1:37 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
... I didn't see the solution for 2D immediately ... but suddenly I did.
Neat puzzle.
On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside.
Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger.
Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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There's a much simpler and cleaner solution to the 2D case. On Fri, Jan 12, 2018 at 11:13 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I was an idiot not to look at the the 2D case first. I don't have a detailed solution, but the sketch is, a roughly square section in the middle cut into planks, which get reassembled end-to-end to make the outside of the square, and then the four remaining sections of the circle get chopped up fine enough to fill in the middle. I would guess it can be done in about 20 pieces at most.
On Fri, Jan 12, 2018 at 1:37 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
... I didn't see the solution for 2D immediately ... but suddenly I did.
Neat puzzle.
On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside.
Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger.
Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Well, you've stumped me! On Fri, Jan 12, 2018 at 2:15 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
There's a much simpler and cleaner solution to the 2D case.
On Fri, Jan 12, 2018 at 11:13 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I was an idiot not to look at the the 2D case first. I don't have a detailed solution, but the sketch is, a roughly square section in the middle cut into planks, which get reassembled end-to-end to make the outside of the square, and then the four remaining sections of the circle get chopped up fine enough to fill in the middle. I would guess it can be done in about 20 pieces at most.
On Fri, Jan 12, 2018 at 1:37 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
... I didn't see the solution for 2D immediately ... but suddenly I did.
Neat puzzle.
On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside.
Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger.
Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Oh. If we are permitting tangency contact, then I see the solution Tom has in mind :) On Fri, Jan 12, 2018 at 2:16 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Well, you've stumped me!
On Fri, Jan 12, 2018 at 2:15 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
There's a much simpler and cleaner solution to the 2D case.
On Fri, Jan 12, 2018 at 11:13 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I was an idiot not to look at the the 2D case first. I don't have a detailed solution, but the sketch is, a roughly square section in the middle cut into planks, which get reassembled end-to-end to make the outside of the square, and then the four remaining sections of the circle get chopped up fine enough to fill in the middle. I would guess it can be done in about 20 pieces at most.
On Fri, Jan 12, 2018 at 1:37 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
... I didn't see the solution for 2D immediately ... but suddenly I did.
Neat puzzle.
On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside.
Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger.
Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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By tangency contact I guess you mean contact with negligible mating surface. If we don't permit that and we want a minimum mating surface, it's easy to change it to a solution requiring only twice the number of pieces. On Fri, Jan 12, 2018 at 11:17 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Oh. If we are permitting tangency contact, then I see the solution Tom has in mind :)
On Fri, Jan 12, 2018 at 2:16 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Well, you've stumped me!
On Fri, Jan 12, 2018 at 2:15 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
There's a much simpler and cleaner solution to the 2D case.
On Fri, Jan 12, 2018 at 11:13 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I was an idiot not to look at the the 2D case first. I don't have a detailed solution, but the sketch is, a roughly square section in the middle cut into planks, which get reassembled end-to-end to make the outside of the square, and then the four remaining sections of the circle get chopped up fine enough to fill in the middle. I would guess it can be done in about 20 pieces at most.
On Fri, Jan 12, 2018 at 1:37 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
... I didn't see the solution for 2D immediately ... but suddenly I did.
Neat puzzle.
On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside.
Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger.
Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Yes, I see that. Cool. On Fri, Jan 12, 2018 at 2:21 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
By tangency contact I guess you mean contact with negligible mating surface.
If we don't permit that and we want a minimum mating surface, it's easy to change it to a solution requiring only twice the number of pieces.
On Fri, Jan 12, 2018 at 11:17 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Oh. If we are permitting tangency contact, then I see the solution Tom has in mind :)
On Fri, Jan 12, 2018 at 2:16 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Well, you've stumped me!
On Fri, Jan 12, 2018 at 2:15 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
There's a much simpler and cleaner solution to the 2D case.
On Fri, Jan 12, 2018 at 11:13 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I was an idiot not to look at the the 2D case first. I don't have a detailed solution, but the sketch is, a roughly square section in the middle cut into planks, which get reassembled end-to-end to make the outside of the square, and then the four remaining sections of the circle get chopped up fine enough to fill in the middle. I would guess it can be done in about 20 pieces at most.
On Fri, Jan 12, 2018 at 1:37 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
... I didn't see the solution for 2D immediately ... but suddenly I did.
Neat puzzle.
On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler < acwacw@gmail.com> wrote:
> It's obviously possible to dissect a sphere into pieces that could be > reassembled into something that looks like a cube from the outside. Inside, > it would be hollow, possibly with a bunch of spare pieces rattling around > inside. > > Feasibility sketch: build thin polyhedral planks from the inner part of the > sphere, with 45-degree bevels where necessary to let them form the cube's > edges and vertices. All the extra material could be chopped up into > manageable chunks and hidden in the interior. If there's too much extra > material, redesign with thinner planks to make the outer cube bigger. > > Intuitively, it feels like we could get away with just a few dozen pieces > or so: maybe 4 to 6 per face, and then an approximately equal number to > pack inside ... but I don't know. Can anybody provide an explicit > construction, or make lower-bound arguments about the number of pieces? > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
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This can, in turn, be optimised into a solution with 25% fewer pieces. -- APG.
Sent: Friday, January 12, 2018 at 7:24 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Sphere to cube dissection?
Yes, I see that. Cool.
On Fri, Jan 12, 2018 at 2:21 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
By tangency contact I guess you mean contact with negligible mating surface.
If we don't permit that and we want a minimum mating surface, it's easy to change it to a solution requiring only twice the number of pieces.
On Fri, Jan 12, 2018 at 11:17 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Oh. If we are permitting tangency contact, then I see the solution Tom has in mind :)
On Fri, Jan 12, 2018 at 2:16 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Well, you've stumped me!
On Fri, Jan 12, 2018 at 2:15 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
There's a much simpler and cleaner solution to the 2D case.
On Fri, Jan 12, 2018 at 11:13 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I was an idiot not to look at the the 2D case first. I don't have a detailed solution, but the sketch is, a roughly square section in the middle cut into planks, which get reassembled end-to-end to make the outside of the square, and then the four remaining sections of the circle get chopped up fine enough to fill in the middle. I would guess it can be done in about 20 pieces at most.
On Fri, Jan 12, 2018 at 1:37 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
> ... I didn't see the solution for 2D immediately ... but suddenly I did. > > Neat puzzle. > > On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler < acwacw@gmail.com> wrote: > > > It's obviously possible to dissect a sphere into pieces that could be > > reassembled into something that looks like a cube from the outside. > Inside, > > it would be hollow, possibly with a bunch of spare pieces rattling around > > inside. > > > > Feasibility sketch: build thin polyhedral planks from the inner part of > the > > sphere, with 45-degree bevels where necessary to let them form the cube's > > edges and vertices. All the extra material could be chopped up into > > manageable chunks and hidden in the interior. If there's too much extra > > material, redesign with thinner planks to make the outer cube bigger. > > > > Intuitively, it feels like we could get away with just a few dozen pieces > > or so: maybe 4 to 6 per face, and then an approximately equal number to > > pack inside ... but I don't know. Can anybody provide an explicit > > construction, or make lower-bound arguments about the number of pieces? > > _______________________________________________ > > math-fun mailing list > > math-fun@mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > > -- > -- http://cube20.org/ -- http://golly.sf.net/ -- > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I see that! Very nice. Funny how knowledge of existence of a solution leads you to see that solution . . . -tom On Fri, Jan 12, 2018 at 12:07 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
This can, in turn, be optimised into a solution with 25% fewer pieces.
-- APG.
Sent: Friday, January 12, 2018 at 7:24 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Sphere to cube dissection?
Yes, I see that. Cool.
On Fri, Jan 12, 2018 at 2:21 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
By tangency contact I guess you mean contact with negligible mating surface.
If we don't permit that and we want a minimum mating surface, it's easy to change it to a solution requiring only twice the number of pieces.
On Fri, Jan 12, 2018 at 11:17 AM, Allan Wechsler <acwacw@gmail.com> wrote:
Oh. If we are permitting tangency contact, then I see the solution Tom has in mind :)
On Fri, Jan 12, 2018 at 2:16 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Well, you've stumped me!
On Fri, Jan 12, 2018 at 2:15 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
There's a much simpler and cleaner solution to the 2D case.
On Fri, Jan 12, 2018 at 11:13 AM, Allan Wechsler < acwacw@gmail.com> wrote:
> I was an idiot not to look at the the 2D case first. I don't have a > detailed solution, but the sketch is, a roughly square section in the > middle cut into planks, which get reassembled end-to-end to make the > outside of the square, and then the four remaining sections of the circle > get chopped up fine enough to fill in the middle. I would guess it can be > done in about 20 pieces at most. > > On Fri, Jan 12, 2018 at 1:37 PM, Tomas Rokicki < rokicki@gmail.com> wrote: > > > ... I didn't see the solution for 2D immediately ... but suddenly I did. > > > > Neat puzzle. > > > > On Fri, Jan 12, 2018 at 10:07 AM, Allan Wechsler < acwacw@gmail.com> > wrote: > > > > > It's obviously possible to dissect a sphere into pieces that could be > > > reassembled into something that looks like a cube from the outside. > > Inside, > > > it would be hollow, possibly with a bunch of spare pieces rattling > around > > > inside. > > > > > > Feasibility sketch: build thin polyhedral planks from the inner part of > > the > > > sphere, with 45-degree bevels where necessary to let them form the > cube's > > > edges and vertices. All the extra material could be chopped up into > > > manageable chunks and hidden in the interior. If there's too much extra > > > material, redesign with thinner planks to make the outer cube bigger. > > > > > > Intuitively, it feels like we could get away with just a few dozen > pieces > > > or so: maybe 4 to 6 per face, and then an approximately equal number to > > > pack inside ... but I don't know. Can anybody provide an explicit > > > construction, or make lower-bound arguments about the number of pieces? > > > _______________________________________________ > > > math-fun mailing list > > > math-fun@mailman.xmission.com > > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math- fun > > > > > > > > > > > -- > > -- http://cube20.org/ -- http://golly.sf.net/ -- > > _______________________________________________ > > math-fun mailing list > > math-fun@mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math- fun > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
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participants (3)
-
Adam P. Goucher -
Allan Wechsler -
Tomas Rokicki