For polygons, does the circumference/"diameter" have similar properties to pi for circles?"

Factual Questions
21

What do you want the circumference in terms of? As noted above, the ratio of the circumference to the outer and inner radii is 2n * sin(a half turn/n) and 2n * tan(a half turn/n) = 2n * sin(a half turn/n)/sqrt(1 - (sin(a half turn/n))[sup]2[/sup]), respectively.

This ties the answer to the sine function, but we can express sin(a half turn/n) in other convenient fashions: when n is a power of 2, we may use repeated applications of the “half-angle” formulae of trigonometry to express this using basic arithmetic and square roots, and more generally, by consideration of the “addition” formulae of trigonometry, we find, as noted above, an integer coefficient polynomial of which this is the smallest root.

22

Wouldn’t it be easier to add the length of each edge?

23

I was wondering if there’s a way to express the circumference relative to the “diameter” without using trig and pi. Just like Pythagoras was able to solve for certain properties of a triangle without needing pi, is there a way to get a similar answer to your equation without using pi and the trig functions?

My original question was wondering if that ratio in polygons had similar properties as pi does to circles. But then when the equations shown for the circumference of polygons all have pi (or trig functions), I can’t help wondering if the use of pi is ‘tainting’ the results. I wonder if an equation to solve for the polygon circumference without using pi (or one of it’s infinite series approximations) would give different results. Since pi and the trig functions are all approximations, perhaps being able to solve for it without using them would give a different set of results.

24

That is what I’m doing. Each edge is 2 * sin(a half turn/n) times as long as the distance of the edge from the center. So if you want the circumference in terms of the inradius, you get n times that.

If all you want is the circumference in terms of the edge length, of course, the answer is simply “The circumference is n times the edge length”. But that’s so trivial I presumed that wasn’t what was being sought.

25

For what it’s worth, I didn’t use pi when expressing polygon perimeters above. It was perhaps unfortunate that I used “n” as the variable indicating the number of sides, when this looks like a pi symbol; that was actually a lower-case N, though.

I did invoke the sine function, but that’s because the answer is what it is, and what it is is a minor transform of the sine function. No way around that. If you understand the sine function at the relevant inputs, you can use that to calculate the polygon perimeter in terms of its inradius, and conversely, if you understand the polygon perimeter in terms of its inradius, you can use that to calculate the sine function at the relevant inputs.

26

It is not possible to express these formulas completely divorced from pi, because these formulas can be used to get pi. Everything in mathematics is related, and the relationships can work either way.

27

For a hexagon, the perimeter divided by the diameter is 3 – the only whole-number “pi” out of all the polygons. Coopers (barrelmakers) used to use this fact to set the width of the dividers (compasses) – if you set it so that you could walk the points around the circumference six times and come back where you started, then your compass points are equal to the radius of the circle that circumscribes the hexagon. It tells you how nig the “lid” for your barrel has to be. I suspect that it was similar reasoning that lead to the famous Old Testament value of pi = 3 when measuring Solomon’s “Molten Sea”

28

Personally, I think that the simplest and most likely explanation for the “biblical value of pi” is that it’s just rounded, and a correct usage of significant figures.

But if you insist on making it literally true in some way, you can point to the fact that it’s bowl-shaped, and assume that the diameter was measured along the surface, as it would be if measured via a measuring wheel.

29

I like the hexagon one that CalMeacham posted.

Not sure if this factoid is the kind of thing you were looking for, filmore, but I offer it nonetheless:

Start with a square. Draw a “diameter” that connects two corners of the square. The relationship of that “diameter” to the square’s “circumference” is that the “diameter’s” length is sqrt2 * circumference/4.

Of course, with a square, “diameter” and “circumference” are not really the proper terms.

30

I was thinking the fastest way to figure an irregular polygon … one kinda wants to be in and out in a hurry in such situations … if you take my meaning …

31

@ Filmore:

It seems to me that your hope is that, if were to reduce the ratio P / D for regular polygons to something that eliminated PI, then we might find some new interesting… something. In particular, we might find something more basic than PI. Now, this will probably sound absurd to the professional mathematician, who accepts PI at “face value”. It exists, it is beautiful, and it is ubiquitous in mathematics.

However, to an amateur mathematician, it might seem that there is “lots of room” in the world of mathematics in the vicinity of PI. After all, it is well known to amateurs and professionals alike that (1) the transcendental numbers are more numerous than the rationals (uncountably infinite as opposed to countably infinite) and that (2) there is, as of this point in time, no algebra of transcendental numbers. Despite their numerousness, they are mysterious.

Now, as I said, I think that these are the sorts of amateur “inklings” that are seen as “crank” inklings by the average professional mathematician. This particular ground is well-explored (VERY). No mere amateur, or forum full of amateurs, is going to find something new, more basic than PI – or so the feeling goes, I think.

But, in truth, who knows? Perhaps it’s worth exploring. I am always interested to follow out ideas, whether or not they prove to be fruitful. It is often by struggling and failing that we come to understand. For my part, I’ve learned new things due to your original question, Filmore. Perhaps I shall learn more in following on after the thoughts which provoked the question.

32

This is really by definition: Transcendental numbers are defined by the fact that they can’t be derived from certain standard algebraic steps. Now, one can certainly construct other algebraic processes that catch some specific transcendentals in their nets: After all, we do in fact do plenty of work with pi and e and a number of others. But for any algebraic system you construct, you’re always going to be missing the vast majority of real numbers.

33

My understanding was that the possibility of an algebra of transcendental numbers was still open. Indeed, it seems so. For example, if Schanuel’s conjecture were to be proven, then (from the article: Schanuel's conjecture - Wikipedia)

“Schanuel’s conjecture, if proved, would also settle the algebraic nature of numbers such as e + PI and e^e, and prove that e and PI are algebraically independent simply by setting z1 = e and z2 = PI * i, and using Euler’s identity.”

In fact, what has happened is that mathematicians have defined transcendental numbers as those numbers which are not closed under the operations of addition, subtraction, multiplication, division, and root extraction over the polynomials with all integer coefficients. True, there is a certain meaning of “algebraic” which “by definition” excludes transcendental numbers, but this meaning doesn’t mitigate against the possibility of an extension of algebra over the transcendentals. It would still be recognizably algrebra if we could add e + PI, or multiples of same.

34

It’s not clear, to me at least, what is meant by the phrase “an algebra of transcendental numbers”.

35

The term “algebra” may have several meanings here. Perhaps that is part of the problem.

The least contentious meaning of algebra might be something like:
(A1) ALGEBRA:
a system of naming for symbols, together with rules for the valid manipulation of same, so that (i) new expressions can be generated and (ii) the validity of novel expressions may be ascertained

This is the purely syntactic view of algebra, as essentially nothing more than a formal system. It is possible to study such systems in their own right, and philosophers, logicians, mathematicians, computer scientists, and others do so.

But of course this is not the professional mathematician’s understanding of full-blooded “algebra”. A useful algebra – an algebra that can do something interesting in mathematics – requires a semantics as well as a syntax.

What constitutes a semantics is not philosophically neutral. I will therefore not apologize for any contentious philosophical claims in what follows. But please correct any factual errors.

Adding syntax to semantics, then, an “algebra” is something like:
(A2) ALGEBRA:

  1. a universe of discourse (things which constants and variables may vary over)
  2. a set (possibly empty) of constants and variables
  3. a set of axioms that assert what are valid ways to combine variables and constants
  4. an interpretation which assigns meanings to the results of expressions derived using the algebra

It may be seen that these is actually a very relaxed definition of “algebra”. In particular, it would accept all of the following as “algebras”:

  1. a monoid
  2. a semigroup
  3. a group
  4. a ring
  5. an integral domain
  6. a module
  7. a field

Essentially, this is the meaning of “algebra” used when a student takes a course in “Modern Algebra”. It is further formalized and made fully general in the field of “Universal Algebra”, which creates “signatures” for, and formalizes, various sets of axioms, allowing us to take algebras themselves as objects.

Finally, there is a third and very important meaning of “algebra”. It is the most substantive meaning of algebra.
(A3) ALGEBRA:
An “algebra” is (loosely speaking) a vector space together with an associative, commutative product on that space. Essentially, in this final definition, we think of “algebra” as what students do throughout their high school, and early college, careers. It is the “algebra” over complex numbers, with its two operations and their inverses (+ and -, * and /) and its additional axioms of closure, associativity, commutativity, distributivity, and identity elements. (I may be missing something…)

With these ideas in mind, I assert that, to say there is an “algebra” of transcendental numbers is to say:
(1) there is a universe of discourse Ut (read: U-sub-t) over transcendental numbers
(2) there is at least one binary operation T over Ut
(3) the operation of T on elements u of Ut yields only further elements of Ut

What more need be required? If we (or mathematicians) could create an “algebra” that consisted of nothing more than a single operation (without inverse) that yielded new transcendental numbers – numbers new to science – then this would be very interesting.

Now, as things stand (from what I can see as a non-professional in this area), it is likely that mathematicians might actually need to relax requirement (3). Many operations on transcendentals yield irrational numbers that are not themselves transcendental. Maybe that’s okay. <shrugs> It would still be a very interesting situation if we had a way to “duck in and out”, so to speak, of the transcendentals in a way that is systematic and productive.

So, in defense of the original poster, maybe there is room for new ideas in the basement where PI and e lurk. Whatever happens, PI and e will themselves continue to exist, but we may find, 'ere all is said and done, that they are mere bits of ice, while the rest of the 'berg floats beneath, giving us a new perspective on their meaning in the fuller mathematical universe.

36

Er, I don’t understand what the significance of this is supposed to be.

It’s trivial to define a structure which consists of the transcendental numbers, along with a binary operation that produces only new transcendental numbers. For example, let the binary operation take any inputs to e, regardless of what the inputs are. Or, the binary operation could take x and y to x + y if this is transcendental, and x + y + e otherwise. Or, least trivially yet, take the decimal expansions of (the fractional components of) x and y, intertwine them to get a sequence d[sub]0[/sub], d[sub]1[/sub], d[sub]2[/sub], … of digits, then construct the sum of d[sub]i[/sub]/10[sup]i![/sup] over all i, which will be a Liouville number and therefore transcendental.

It’s not hard to contrive operations which construct numbers which are manifestly transcendental. The hard thing is to demonstrate that numbers specified using familiar old-hat arithmetic operations are transcendental.

None of this makes π all that mysterious, mind you. Sure, it’s unknown whether π + e is transcendental. So what? It’s also unknown whether 2^e is transcendental, but no one considers 2 all that mysterious.

37

Though, to be fair, inverses would be messy in that system.

38

Well, bodhisattvaforfhire did say “a single operation (without inverse)”.

39

@ Indistinguishable,

You point out (correctly) that it is easy to produce trivial cases of an algebra of transcendental numbers in exactly the sense of “algebra” that I gave in my recent post. Now, you are surely correct that we should avoid regarding as “significant mathematics” any operation which is merely the application of known results in the service of trivial consequences of those results. (An infinite pile of trivial results is still… trivial.) Good point!

In short, I answered the literal meaning of your challenge, but I failed its spirit. In particular, I provided a definition for “algebra of transcendentals” which showed that such exists, but it did not show that such an algebra is significant. For example, I failed to show that “good ol’” [my scare quotes] addition is the operation in the “algebra” that I presented. Another good point! (Among other failures, I did not prove Schaunel’s Conjecture, or any other theorem which would give true significance to the idea of an “algebra of transcendentals”. For my part, I can live with this failure.)

Still, we are making progress, in my opinion. You have now stated clearly that it is possible to create an algebra over the transcendentals. It is not mere nonsense to assert so. It does have a clear meaning in mathematics. (I call explicit attention to this only because an uninformed, or un-generous, reader of your prior remark might have been deluded into thinking that you did not see any significance to the “algebra of transcendentals” because you thought such an algebra was not possible. Plainly, you do think it is possible, since you produced on the spot an algorithm for producing trivial results of a trivial algebra over the transcendentals. I applaud your mathematical acumen, for my part. I cannot duplicate it.)

As you also plainly say, it is not even that an “algebra of transcendentals” would not be significant as an achievement in mathematics, if (at the least) it involved “good ol’” addition. For example, proving PI + e would be significant at least in the sense that it would involve proving something that is difficult to prove. This is surely not “trivial”.

But what about your final statements, namely, an argument to the effect that there is “nothing mysterious” about PI…? As I see things, you provided a “straw person” criterion for PI’s being mysterious, then proceeded to knock down the criterion with an analogy to 2^e. But, clearly, it is not true that PI is mysterious only if the transcendental status of PI + e is unknown. Couldn’t there be other reasons why PI might be mysterious? Put another way, if your criterion were the only reason that PI could be mysterious, I wouldn’t believe it to be mysterious, either.

In fact, I pointed out that, possibly, there is an entire algebra of transcendentals that one might use if only one could prove (for example) the Schaunel Conjecture. PI is mysterious (at least) because of the fact that oodles of mathematicians try yet fail to prove that it is one of an (uncountably) infinite number of objects involved in a non-trivial algebra over such numbers.

An algebra of transcendentals (using good ol’ addition) would be “non-trivial” in at least two ways: (1) if it exists, it is not obvious as to how to construct it and (2) if it exists, it may give mathematicians the tools necessary to prove the transcendental status of a host (finite? infinite? uncountably infinite?) of numbers. Since transcendental numbers are not amenable to statement using well-developed tools like the algebra of polynomial equations with integer coefficients, this could be used systematically to steer mathematical efforts wherever those numbers were encountered in mathematics (among other possible benefits). One would not need to doubt whether one was in “transcendental land”, but instead would know. In short, knowing easily whether a given number were transcendental could be useful, and such an algebra would give one a much better starting point for establishing, for a given number, whether it had such a status. (That the algebra encompassed all transcendental numbers would be something like the “fundamental theorem of transcendental numbers”. But that is certainly more than Schanuel’s Conjecture states, or that I have maintained is possible. Nor have I heard or seen anyone conjecture such a thing.)

To return to the more general point in this discussion, for all I know, thinking about the relationship of regular polygons to circles might be among the interesting ways of approaching the Schaunel Conjecture. So, Filmore’s intuition could be correct, even though he is at least as much of an amateur mathematician as I am.

40

Again, I don’t know what you mean by the phrase you keep using. What does “An algebra of transcendentals (using good ol’ addition)” mean?

It’s manifestly the case that a transcendental added to another transcendental may result in a non-transcendental (e.g., e + (3 - e)). It’s also easy enough to produce a subset of the transcendentals which is closed under binary addition (e.g., the set of positive integer multiples of e).

Are you just looking for an easy way to determine whether expressions written using ordinary arithmetic operations and ordinary constants like e and π come out transcendental or not? This would indeed be a breakthrough.

I still don’t consider any of this to make π “mysterious”. This seems a kind of mysticism which infects pop mathematics which I am not fond of. There are things we understand very well about π. There are also things we don’t understand so well which invoke π. The same is true of 2 (is the Collatz Conjecture (involving division by 2) true? Is the Goldbach Conjecture (involving multiples of 2, as well as 2 primes) true? Is sqrt(2) a normal number in any or all bases?) and everything else.