Comments on: Can You Solve “Unsolvable” Mathematical Problems? (Nov, 1931)

By: Zyzzyva

Zyzzyva — Tue, 15 Dec 2009 04:23:47 +0000

It’s insoluble. It had been proved insoluble by then – IIRC, more than fifty years before. Callahan was (provably) a crank.

That said, the article does get one thing right: squaring the circle has probably caused exactly “as many headaches among mathematicians as the perpetual motion problem has among inventors”, eg, none, if you don’t count the headache of fending off the cranks.

By: Firebrand38

Firebrand38 — Wed, 07 Oct 2009 17:42:56 +0000

Maaz: Don’t just make an unsubstantiated statement. Solve the problems.

By: Maaz

Maaz — Wed, 07 Oct 2009 17:40:52 +0000

There are no unsolvable problems…Maybe the problem is non existant in nature.

By: George Trudeau

George Trudeau — Mon, 08 Dec 2008 19:27:11 +0000

I’ve discovered a remarkable solution to this problem, unfortunately this text box is too small to hold it. P. de Fermat

By: fluffy

fluffy — Sat, 06 Dec 2008 20:37:47 +0000

Yeah, I’m on a Mac, which has plenty of mnemonic shortcuts for actual symbols, but a superscript is not semantically a symbol. I could just use the Unicode character entry palette, but writing proper HTML is usually a lot faster.

π is just π. It’s a standard HTML entity.

By: Matt

Matt — Sat, 06 Dec 2008 19:53:23 +0000

For superscript 2, hold down the alt key, type 0178 on your numeric keypad (num lock must be on), and release the alt key. Presto ² shows up. Works for all the standard ASCII characters on PCs. You can look up the codes with the Character Map in Windows. I don’t know if Macs have a similar shortcut.

By: Toronto

Toronto — Sat, 06 Dec 2008 04:22:45 +0000

Fluffy – I’m impressed you got a pi-like character to show up, never mind the superscript.

I have to roll the platen back half a click manually via the right hand roller handle to do that on this typewriter…

By: fluffy

fluffy — Fri, 05 Dec 2008 20:56:42 +0000

Obviously that should be πr^2. This comment system eats ^tags.

By: fluffy

fluffy — Fri, 05 Dec 2008 20:55:50 +0000

I believe that trisecting an angle via plane geometry has been proven impossible by contradiction (i.e. presupposing that it’s possible, showing that the math falls apart). However, trisecting in plane geometry is possible via folding, which is a rather neat modern branch of geometry which has grown out of origami.

The article also mis-states what squaring the circle is about. It’s not about finding the precise area of a circle mathematically, but geometrically constructing a square with the same area as any given circle. Given that the area of a circle is πr2 and the area of a square is r2, that means that you’d have to somehow construct a square whose side’s length is precisely √pi of the circle’s radius — not a feasible construction! (And, again, probably disprovable by contradiction.)

By: Buddy

Buddy — Fri, 05 Dec 2008 11:08:04 +0000

It is possible to bisect an angle, and by recursive bisections, 4-sect, 8-sect, 16-sect, etc. an angle.

Using the limit:

1/3 = 1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + …

Multiplying by x:

x/3 = x/2 – x/4 + x/8 – x/16 + x/32 – x/64 + …

It is evident that x/2, x/4, x/8, x/16, x/32, x/64, etc. are equivalent to recursive bisections of x.

Hence is possible to trisect an angle x to any finite precision using recursive bisections in a finite number of steps.

It doesn’t count as a proof because in theory, an infinite number of steps would be required to trisect exactly.

The limit can also be expressed 1/3 = 1/4 + 1/16 + 1/64 + … + 1/(n**4).

It’s the same for all n not equal to -1, 0, 1, even non-integer.

1/2 = 1/3 + 1/9 + 1/27 + …
1/3 = 1/4 + 1/16 + 1/64 + … = 1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + …
1/4 = 1/5 + 1/25 + 1/125 + … = 1/3 – 1/9 + 1/27 – 1/71 + …
…

Any fractions which can be expressed in sums of powers of two are amenable to this technique:

1/3 = 1/4 + 1/16 + 1/64 + …
1/5 = 1/4 – 1/16 + 1/64 – …
1/7 = 1/8 + 1/64 + 1/512 + …
1/9 = 1/8 – 1/64 + 1/512 – …
1/15 = 1/16 + 1/256 + 1/4096 + …
1/17 = 1/16 – 1/256 + 1/4096 – …

Meaning it’s possible to trisect, or 5-, 7-, 9-, 15-, 17-sect, etc. any angle to any finite precision in finitely many steps.