OK, I had dropped this for a while, but here are my thoughts. I haven’t scrubbed everything that could be seen through rot13 because it became excessively unreadable
For Part 1: gur enqvhf bs gur pragre fcurer vf gur qvfgnapr orgjrra bar bs gur qvnzrgre-1/2 fcurerf naq gur pragre.
Gur qvfgnapr sebz gur pragre bs gur fvqr-fcurer gb gur pragre bs gur birenyy phor vf fdeg(A)/4. Fhogenpg bss n dhnegre sbe gur enqvhf bs gur fcurer, naq jr unir gur enqvhf bs gur pragre fcurer: (fdeg(A)-1)/4. Guvf jvyy xvff gur bhgfvqr bs gur fvqr-1 ulcrephor jura gung’f rdhny gb n unys, juvpu unccraf ng avar qvzrafvbaf. Zber guna gung naq vg jvyy rkgraq bhgfvqr.
Part 2: I admit that I didn’t have the volume of high-dimensional spheres memorized, but it’s up on wikipedia, and from there it’s just a matter of graphing and seeing where the curve crosses 1, taking into account the radius formula derived above.. I haven’t done it, but will eventually.
Qvfgnapr sebz prager bs phor gb prager bs “pbeare” fcurer rdhnyf fdeg(a) gvzrf qvfgnapr ba bar nkvf = fdeg(a) bire sbhe. Enqvhf bs “pbeare” fcurer rdhnyf bar bire sbhe. Gurersber enqvhf bs prageny fcurer = (fdeg(a) zvahf bar) bire sbhe. Bs pbhefr guvf trgf nf ynetr nf lbh cyrnfr sbe ynetr a. Vg rdhnyf bar unys, sbe n qvnzrgre bs bar, jura (fdeg(a) zvahf bar) bire sbhe rdhnyf bar unys ⇔ fdeg(a) zvahf bar rdhnyf gjb ⇔ fdeg(a) rdhnyf guerr ⇔ a rdhnyf avar.
dhrfgvba gjb
Guvf arire unccraf. Hfvat Fgveyvat’f sbezhyn jr svaq gung gur nflzcgbgvpf ner abg snibhenoyr, naq vg’f rnfl gb pbzchgr gur svefg ubjrire-znal inyhrf ahzrevpnyyl. V unira’g gebhoyrq gb znxr na npghny cebbs ol hfvat rkcyvpvg obhaqf rireljurer, ohg vg jbhyq or cnvashy engure guna qvssvphyg.
dhrfgvba guerr
Abar. Lbh pnaabg rira svg n ulcrefcurer bs qvnzrgre gjb orgjrra gjb ulcrecynarf ng qvfgnapr bar, naq gur ulcrephor vf gur vagrefrpgvba bs bar uhaqerq fcnprf bs guvf fbeg.
Oooh, I dropped a factor of 2 in the second one and didn’t notice because it takes longer than you’d expect before the numbers start increasing. Revised answer:
dhrfgvba gjb
Vs lbh qb gur nflzcgbgvpf pbeerpgyl engure guna jebatyl, gur ibyhzr tbrf hc yvxr (cv gvzrf r bire rvtug) gb gur cbjre a/2 qvivqrq ol gur fdhner ebbg bs a. Gur “zvahf bar” va gur sbezhyn sbe gur enqvhf zrnaf gung gur nflzcgbgvp tebjgu gnxrf ybatre gb znavsrfg guna lbh zvtug rkcrpg. Gur nafjre gb gur dhrfgvba gheaf bhg gb or bar gubhfnaq gjb uhaqerq naq fvk, naq V qb abg oryvrir gurer vf nal srnfvoyr jnl gb trg vg bgure guna npghny pnyphyngvba.
On the face of it, the premise seems wrong. For any finite number of dimensions, there will be a finite number of objects in the cube, which means you aren’t getting any infinity shenanigans—it’s just high-dimensional geometry. And in no non-shenanigans case will the hypervolume of a thing be greater than a thing it is entirely inside of.
Another math problem:
https://protokol2020.wordpress.com/2017/01/11/and-yet-another-geometry-problem/
OK, I had dropped this for a while, but here are my thoughts. I haven’t scrubbed everything that could be seen through rot13 because it became excessively unreadable
For Part 1: gur enqvhf bs gur pragre fcurer vf gur qvfgnapr orgjrra bar bs gur qvnzrgre-1/2 fcurerf naq gur pragre.
Gur qvfgnapr sebz gur pragre bs gur fvqr-fcurer gb gur pragre bs gur birenyy phor vf fdeg(A)/4. Fhogenpg bss n dhnegre sbe gur enqvhf bs gur fcurer, naq jr unir gur enqvhf bs gur pragre fcurer: (fdeg(A)-1)/4. Guvf jvyy xvff gur bhgfvqr bs gur fvqr-1 ulcrephor jura gung’f rdhny gb n unys, juvpu unccraf ng avar qvzrafvbaf. Zber guna gung naq vg jvyy rkgraq bhgfvqr.
Part 2: I admit that I didn’t have the volume of high-dimensional spheres memorized, but it’s up on wikipedia, and from there it’s just a matter of graphing and seeing where the curve crosses 1, taking into account the radius formula derived above.. I haven’t done it, but will eventually.
Part 3 looks harder and I’ll look at it later.
Part 1 is good.
dhrfgvba bar
Qvfgnapr sebz prager bs phor gb prager bs “pbeare” fcurer rdhnyf fdeg(a) gvzrf qvfgnapr ba bar nkvf = fdeg(a) bire sbhe. Enqvhf bs “pbeare” fcurer rdhnyf bar bire sbhe. Gurersber enqvhf bs prageny fcurer = (fdeg(a) zvahf bar) bire sbhe. Bs pbhefr guvf trgf nf ynetr nf lbh cyrnfr sbe ynetr a. Vg rdhnyf bar unys, sbe n qvnzrgre bs bar, jura (fdeg(a) zvahf bar) bire sbhe rdhnyf bar unys ⇔ fdeg(a) zvahf bar rdhnyf gjb ⇔ fdeg(a) rdhnyf guerr ⇔ a rdhnyf avar.
dhrfgvba gjb
Guvf arire unccraf. Hfvat Fgveyvat’f sbezhyn jr svaq gung gur nflzcgbgvpf ner abg snibhenoyr, naq vg’f rnfl gb pbzchgr gur svefg ubjrire-znal inyhrf ahzrevpnyyl. V unira’g gebhoyrq gb znxr na npghny cebbs ol hfvat rkcyvpvg obhaqf rireljurer, ohg vg jbhyq or cnvashy engure guna qvssvphyg.
dhrfgvba guerr
Abar. Lbh pnaabg rira svg n ulcrefcurer bs qvnzrgre gjb orgjrra gjb ulcrecynarf ng qvfgnapr bar, naq gur ulcrephor vf gur vagrefrpgvba bs bar uhaqerq fcnprf bs guvf fbeg.
One: Correct
Two: Incorrect
Three: Correct
Oooh, I dropped a factor of 2 in the second one and didn’t notice because it takes longer than you’d expect before the numbers start increasing. Revised answer:
dhrfgvba gjb
Vs lbh qb gur nflzcgbgvpf pbeerpgyl engure guna jebatyl, gur ibyhzr tbrf hc yvxr (cv gvzrf r bire rvtug) gb gur cbjre a/2 qvivqrq ol gur fdhner ebbg bs a. Gur “zvahf bar” va gur sbezhyn sbe gur enqvhf zrnaf gung gur nflzcgbgvp tebjgu gnxrf ybatre gb znavsrfg guna lbh zvtug rkcrpg. Gur nafjre gb gur dhrfgvba gheaf bhg gb or bar gubhfnaq gjb uhaqerq naq fvk, naq V qb abg oryvrir gurer vf nal srnfvoyr jnl gb trg vg bgure guna npghny pnyphyngvba.
Correct.
I gave some Haskell code as a comment over there on my blog, under the posted problem.
1206 dimension is the smallest number. One can experiment with other values.
On the face of it, the premise seems wrong. For any finite number of dimensions, there will be a finite number of objects in the cube, which means you aren’t getting any infinity shenanigans—it’s just high-dimensional geometry. And in no non-shenanigans case will the hypervolume of a thing be greater than a thing it is entirely inside of.
Are you sure, it’s entirely inside?
OK, that’s an angle (pun intended) I didn’t catch upon first consideration.
High-dimensional cubes are really thin and spiky.
They are counterintuitive. A lot is counterintuitive in higher dimensions. Especially something, I may write about in the future.
This 1206 business is even Googleable. Which I have learned only after I have calculated the actual number 1206.
https://sbseminar.wordpress.com/2007/07/21/spheres-in-higher-dimensions/