[math-fun] Lifted from Littlewood, hacked from Hacking
I had seen this before, long ago, but then forgotten it. A square can be dissected into finitely many unequal squares, but a cube cannot be dissected into finitely many unequal cubes. Why not?
I remember how the proof starts: Look at the bottom face of the cube and how it's dissected into squares. Jim Propp On Tuesday, February 7, 2017, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I had seen this before, long ago, but then forgotten it.
A square can be dissected into finitely many unequal squares, but a cube cannot be dissected into finitely many unequal cubes.
Why not?
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It then continues: The *smallest* square of that dissection is the bottom of a cube, whose top is a square platform with a nonzero "curb" all the way around its edges. On Tue, Feb 7, 2017 at 9:04 AM, James Propp <jamespropp@gmail.com> wrote:
I remember how the proof starts: Look at the bottom face of the cube and how it's dissected into squares.
Jim Propp
On Tuesday, February 7, 2017, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I had seen this before, long ago, but then forgotten it.
A square can be dissected into finitely many unequal squares, but a cube cannot be dissected into finitely many unequal cubes.
Why not?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Ok, so stack yet smaller cubes on top of it to climb the curb... But then the small cube's top face is dissected into yet smaller squares, and the cube sitting on that square has a curb. Infinite descent, ergo no smallest cube. Hence a cube cannot be tiled with a finite number of distinct cubes.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Tuesday, February 07, 2017 9:16 AM To: math-fun Subject: Re: [math-fun] Lifted from Littlewood, hacked from Hacking
It then continues: The *smallest* square of that dissection is the bottom of a cube, whose top is a square platform with a nonzero "curb" all the way around its edges.
On Tue, Feb 7, 2017 at 9:04 AM, James Propp <jamespropp@gmail.com> wrote:
I remember how the proof starts: Look at the bottom face of the cube and how it's dissected into squares.
Jim Propp
On Tuesday, February 7, 2017, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I had seen this before, long ago, but then forgotten it.
A square can be dissected into finitely many unequal squares, but a cube cannot be dissected into finitely many unequal cubes.
Why not?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> 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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Are there any natural (or at least pretty) ways to tile a cube with an infinite number of distinct cubes? Jim On Tue, Feb 7, 2017 at 8:50 PM, David Wilson <davidwwilson@comcast.net> wrote:
Ok, so stack yet smaller cubes on top of it to climb the curb... But then the small cube's top face is dissected into yet smaller squares, and the cube sitting on that square has a curb. Infinite descent, ergo no smallest cube. Hence a cube cannot be tiled with a finite number of distinct cubes.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Tuesday, February 07, 2017 9:16 AM To: math-fun Subject: Re: [math-fun] Lifted from Littlewood, hacked from Hacking
It then continues: The *smallest* square of that dissection is the bottom of a cube, whose top is a square platform with a nonzero "curb" all the way around its edges.
On Tue, Feb 7, 2017 at 9:04 AM, James Propp <jamespropp@gmail.com> wrote:
I remember how the proof starts: Look at the bottom face of the cube and how it's dissected into squares.
Jim Propp
On Tuesday, February 7, 2017, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I had seen this before, long ago, but then forgotten it.
A square can be dissected into finitely many unequal squares, but a cube cannot be dissected into finitely many unequal cubes.
Why not?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Followup question: Dissect a cube into smaller cubes, with the smallest number of duplicates. Opening bid: Split a 2^3 into eight 1^3's. Duplicity = 7. Do I hear 6? Different direction, going negative: Suppose we allow negative cubes -- negative cubes are simply subtracted after all the positive cubes are placed; neg-cubes erase overlaps of the positive cubes, ideally leaving the target cube covered exactly once. This could also make sense with squares in two dimensions. However, IIRC, the original history article for squared squares mentions an episode where one of their graph-diagrams had negative edges, and they found a simple mechanical tweak to make everything positive. Rich ------ Quoting David Wilson <davidwwilson@comcast.net>:
Ok, so stack yet smaller cubes on top of it to climb the curb... But then the small cube's top face is dissected into yet smaller squares, and the cube sitting on that square has a curb. Infinite descent, ergo no smallest cube. Hence a cube cannot be tiled with a finite number of distinct cubes.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Tuesday, February 07, 2017 9:16 AM To: math-fun Subject: Re: [math-fun] Lifted from Littlewood, hacked from Hacking
It then continues: The *smallest* square of that dissection is the bottom of a cube, whose top is a square platform with a nonzero "curb" all the way around its edges.
participants (5)
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Allan Wechsler -
David Wilson -
Fred Lunnon -
James Propp -
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