# Thinking Outside the Plane

October 16, 2019 12:07 PM Subscribe

Tarski's Plank Problem

... the third dimension. Regarding our original circle as the shadow of a sphere, each strip becomes the shadow of a curved band, like the zone between two latitude lines on a globe. Thanks to a theorem of Archimedes (which was commemorated on his tombstone), these bands have the same surface area no matter where on the sphere they are located, each one covering exactly 1/N of the sphere. Thus it takes at least N bands to cover the sphere, and N planks to cover the original circle.

Adding a third dimension to solve a problem in plane geometry is an unusual trick, but it isn't unprecedented:

Monge's Theorem • Three Equal Circles • Four Travelers Problem

—

^{1,2}asks for the least number of strips of width 1 that will cover a circular hole of diameter N. Though the strips are allowed to overlap, the configuration that will occur to most people is N parallel strips side by side, with no overlap and no gaps. But is this the best possible? The middle strips in this configuration cover the most area, while the strips on the edge don't seem to pull their weight. Could a bunch of strips crossing the center in different directions do better? But the outer circumference also needs to be fully covered, and that's one thing "edge" strips do well. There's no obvious right way to measure each strip's contribution. It's a maddening little puzzle, whose resolution surprisingly lies in...... the third dimension. Regarding our original circle as the shadow of a sphere, each strip becomes the shadow of a curved band, like the zone between two latitude lines on a globe. Thanks to a theorem of Archimedes (which was commemorated on his tombstone), these bands have the same surface area no matter where on the sphere they are located, each one covering exactly 1/N of the sphere. Thus it takes at least N bands to cover the sphere, and N planks to cover the original circle.

Adding a third dimension to solve a problem in plane geometry is an unusual trick, but it isn't unprecedented:

Monge's Theorem • Three Equal Circles • Four Travelers Problem

—

^{1}Tarski's actual question was quite a bit more general, concerning convex bodies in any number of dimensions. By the way, yes, it's that Tarski.

^{2}It has been pointed out that overlapping planks don't cover a hole very well. Perhaps a better way to frame the problem in 2019 is: Given a Bic Wite-Out roller that dispenses a 1 cm-wide tape, how many strokes does it take to cover a circular blot of diameter N?

LOVING all the math posts recently! (Despite, assuredly, reaching only a marginal understanding of most of them.) Neat problem which I had never encountered before; thanks for taking the time to construct such a comprehensive overview. Now for your next post: please share with the class which problems in the third dimension are solved by adding a fourth?

posted by youarenothere at 1:29 PM on October 16, 2019 [3 favorites]

posted by youarenothere at 1:29 PM on October 16, 2019 [3 favorites]

I am delighted by the post, by its firstness, and by the huge wait.

posted by cortex at 1:38 PM on October 16, 2019 [5 favorites]

posted by cortex at 1:38 PM on October 16, 2019 [5 favorites]

As a person who has done a lot of sewing and some quilting, my intuition immediately said this was not possible and I dove into your articles hoping to be shown a trick I had not thought of. Nope. Oh, well.

posted by elizilla at 2:01 PM on October 16, 2019 [1 favorite]

posted by elizilla at 2:01 PM on October 16, 2019 [1 favorite]

Planking is so 90s.

posted by Splunge at 2:18 PM on October 16, 2019 [1 favorite]

posted by Splunge at 2:18 PM on October 16, 2019 [1 favorite]

Great post.

There is a more intuitive 3D-invoking proof of Monge's theorem which involves imagining 3 spheres of different sizes sitting on a plane, then a plane parallel to the first plane descending until it touches the first sphere, at which it tilts along that sphere until it touches the second, then along the axis established by the touching points of the two spheres until it touches the third.

Then the descending plane will clearly be tangent to three cones defined by the spheres taken two at a time, and the line of the intersection of the descended plane with the original plane will contain the apexes of all three cones, which establishes the theorem.

I believe that version of the proof is attributed to a 'Professor Sweet' of Yale, but I couldn't Google it.

posted by jamjam at 2:29 PM on October 16, 2019 [2 favorites]

There is a more intuitive 3D-invoking proof of Monge's theorem which involves imagining 3 spheres of different sizes sitting on a plane, then a plane parallel to the first plane descending until it touches the first sphere, at which it tilts along that sphere until it touches the second, then along the axis established by the touching points of the two spheres until it touches the third.

Then the descending plane will clearly be tangent to three cones defined by the spheres taken two at a time, and the line of the intersection of the descended plane with the original plane will contain the apexes of all three cones, which establishes the theorem.

I believe that version of the proof is attributed to a 'Professor Sweet' of Yale, but I couldn't Google it.

posted by jamjam at 2:29 PM on October 16, 2019 [2 favorites]

This is a wonderful first post aws17576.

posted by unearthed at 2:35 PM on October 16, 2019 [1 favorite]

posted by unearthed at 2:35 PM on October 16, 2019 [1 favorite]

Thank you! This was a great post

posted by PMdixon at 3:50 PM on October 16, 2019 [1 favorite]

posted by PMdixon at 3:50 PM on October 16, 2019 [1 favorite]

Aw, thanks for the niceness, y'all!

Good intuition! Alas, a big part of the business of mathematics is to confirm the expected, and the ability to get excited about that is definitely an acquired taste.

For those who would rather play with an optimization problem that

There's plenty of room to keep pushing on the problem. Per the same Wikipedia article:

What do you think I am, some kind of Einstein?

posted by aws17576 at 5:25 PM on October 16, 2019 [5 favorites]

*As a person who has done a lot of sewing and some quilting, my intuition immediately said this was not possible and I dove into your articles hoping to be shown a trick I had not thought of. Nope. Oh, well.*Good intuition! Alas, a big part of the business of mathematics is to confirm the expected, and the ability to get excited about that is definitely an acquired taste.

For those who would rather play with an optimization problem that

*doesn't*have a clear candidate for the answer, I present tetrahedron packing.In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel. These packings were also the basis of a slightly improved packing obtained by Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%, and by Chen, Engel, and Glotzer in early 2010 with a packing fraction of 85.63%.I was in grad school with the Chen mentioned above when she discovered her packing; she said she got some of her insight into the problem by shaking jars of d4's.

There's plenty of room to keep pushing on the problem. Per the same Wikipedia article:

Tetrahedra do not tile space, and an upper bound below 100% (namely, 1 − (2.6...)·10^{−25}) has been reported.

*Now for your next post: please share with the class which problems in the third dimension are solved by adding a fourth?*What do you think I am, some kind of Einstein?

posted by aws17576 at 5:25 PM on October 16, 2019 [5 favorites]

*I am delighted by the post, by its firstness, and by the huge wait.*

I'm glad it fills a much-needed gap!

posted by aws17576 at 5:25 PM on October 16, 2019 [3 favorites]

*For those who would rather play with an optimization problem that doesn't have a clear candidate for the answer, I present tetrahedron packing.*

Oh, now that's a tempting mess to get into. Makes me wish again that I knew how to use a decent toolset for mucking around with 3D objects experimentally. But I got futzing around with with packing heptagons on a plane last year for a morning and even that was kind of at my practical limit in terms of skillset here, so packing volumes would probably get dizzying quickly.

posted by cortex at 6:20 PM on October 16, 2019

Thank you for this post! I particularly like the Four Travelers problem -- it's very interesting how a rather simple reframing of the problem, treating time as a third dimension, suddenly makes the solution obvious.

posted by Tau Wedel at 3:41 AM on October 17, 2019 [1 favorite]

posted by Tau Wedel at 3:41 AM on October 17, 2019 [1 favorite]

*Alas, a big part of the business of mathematics is to confirm the expected, and the ability to get excited about that is definitely an acquired taste.*

Well tbf in this case you have "person whose name is probably most commonly associated with one of the classic demonstrations that the axiom of choice renders physical intuition meaningless" and "cute projective technique" combined in confirming the expected, so that helps.

posted by PMdixon at 8:09 AM on October 17, 2019

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posted by aws17576 at 12:07 PM on October 16, 2019 [19 favorites]