Wikipedia:Reference desk/Archives/Mathematics/2021 December 22

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December 22

Rotation of a cube on its vertex

A solid of revolution is formed by rotating a shape around an axis so that the resulting figure has cylindrical symmetry. A cube is rotated around an axis that can be defined as a line passing through any vertex and the vertex directly opposite. What solid is formed via this rotation? 172.112.210.32 (talk) 02:04, 22 December 2021 (UTC)

I'd be surprised if it had a name, but it would consist of cones at either end and a Hyperboloid of one sheet in the middle. --RDBury (talk) 05:38, 22 December 2021 (UTC)

Volume calculation of solid is at https://math.stackexchange.com/questions/115743/question-about-a-rotating-cube/115797 Result for a cube of edge 1 is Pi/sqrt(3). -- 10:30, 22 December 2021 Naraht

An animated picture can be seen at https://www.pinterest.co.uk/pin/787426316073761880/ -- SGBailey (talk) 15:47, 22 December 2021 (UTC)

The very word "primorial"

I hesitated whether my question (as follows) should be posted, at the Reference desk/Language, rather than here. Finally, I decided to post it here, because I think mathematicians are more likely to be able to answer it.

The word "factorial", coined at the end of the 18th century, was chosen to mean: the product of all "factors" (not larger than a given natural number). On the other hand: the new word "primorial", inspired by the word "factorial" - but only coined about two centuries later, was chosen to mean: the product of all "primes" (not larger than a given prime).

The word "factorial" was received from the word "factor", by the formula: X+"ial". Using this formula, the new word - received from "prime" - should have been "primial", shouldn't it? On the other hand, the word "primORial" seems to be too artificial, and can't be justified, can it? The only problem in "primial" is its similarity to the word "primal", but in my view - this slight problem doesn't justify replacing "primial" by "primorial". The only language that may have a serious problem with "primial" is Corsican, in which "primial" means "prime", but maybe this is another justification for using the word "primial" as an English word whose meaning is related to the primes.

HOTmag (talk) 09:07, 22 December 2021 (UTC)

@HOTmag: "Primorial" is a portmanteau of "prime" and "factorial". I assume it deliberately used a large part of factorial to sound more like it. It would be harder to guess a connection between "primial" and factorial. I actually wrote a prime number paper with Harvey Dubner who coined it but he is dead so I cannot ask him. Language doesn't have to be logical, and often isn't. See List of "-gate" scandals and controversies for a long list of words made up to sound like something well-known. PrimeHunter (talk) 09:57, 22 December 2021‎ (UTC)

Sure, it is somewhat irregularly formed. But so is the term factorial ; the regular formation in Latin would have been factor + -alis = factoralis, which would have given English ^✽factoral; compare Latin pectoralis. If the Romans had invented a word ^✽factorialis, it would have been derived from factorium (oil press) + -alis. The intrusive -i- was inserted by the Frenchies, who generalized the ending -iel seen on words derived from Latin words with a thematic i, as in artificium + -alis = artificialis → artificiel, to a more freely applied ending – without applying for a permit for this innovation. English factorial was borrowed from French factoriel. So the regular formation from prime without intervening derivational innovations would have been primal, not primial. Forming new words based on suffixes obtained from false (unetymological) splittings is a common thing, such as rebracketing alcohol + -ic to alco + -holic, using the newfangled suffix to create work + -a- + -holic = workaholic. (Some contributors here are primaholics.) Other productive suffixes obtained from unetymological splittings are -athon and -burger.  --Lambiam 10:11, 22 December 2021 (UTC)

I'm not sure what you mean by primaholics (me?) but some contributors are Wikipedia:Wikipediholic. PrimeHunter (talk) 10:36, 22 December 2021 (UTC)

Oh, interesting. But please notice, that both "factorial" and "primial" are legitimate English constructions (we disregarding semantics), because the construction "factorial" may be regarded as derived from the English word "factory" + "al", whereas the construction "primial" may be regarded as derived from the (Shakespearean) English word "primy" + "al". Indeed, "primy" is an adjective rather than a noun, but the suffix "al" may be added (yet rarely) to adjectives as well. HOTmag (talk) 10:39, 22 December 2021 (UTC)

Thankxs (which I think can't be regarded as a portmanteau, even though it may originally have meant "thank ax", or "think Xmas, I'm not sure. Anyways, isn't the X in "thankxs" redundant? Probably it is, but who cares?). Do you think there's any way to inquire when he coined "primorial"? If you know when, you may want to add this piece of information to our article primorial. HOTmag (talk) 10:39, 22 December 2021 (UTC)

Referenced in Primorial § References:

• Dubner, Harvey (1987). "Factorial and primorial primes". J. Recr. Math. 19: 197–203.

 --Lambiam 21:25, 22 December 2021 (UTC)

Fascinating discussion. It is probably not that relevant to note that the definition of factorial given is not accurate: it should be "the product of all positive integers less than or equal to n". Is there a word for the product of all factors of n? In cases where no factor occurs with a power greater than 1, this number will be n (1.2.5=10). I can think of no particular use for the number so defined, which is why it may not have attracted a special name, but the phrase "no particular use" will of course appeal to pure mathematicians! --Verbarson ^talk_edits 22:22, 22 December 2021 (UTC)

Would it be radical to call this its "radical"?  --Lambiam 23:47, 22 December 2021 (UTC)

Thank you Lambian; since it has a name, I assume it is a more useful way of interpreting the words. I was thinking that 12 -> 1.2.3.4.6 = 72 (analogous to perfect numbers, where non-prime factors are included), whereas radical(12) = 2.3 = 6 --Verbarson ^talk_edits 00:15, 23 December 2021 (UTC)

The common term for all prime and non-prime factors is divisors, or proper divisors if you exclude the number itself. OEIS:A007955 is "Product of divisors of n." A comment says: Sometimes called the "divisorial" of n. OEIS:A007956 is "Product of proper divisors of n." No name is given. PrimeHunter (talk) 02:33, 23 December 2021 (UTC)

A007956(12) = 144 since 1×2×3×4×6 = 144. PrimeHunter (talk) 02:36, 23 December 2021 (UTC)

<embarrassment> --Verbarson ^talk_edits 19:04, 23 December 2021 (UTC)

Today I learned that [[OEIS:A007956]] syntax exists. —Tamfang (talk) 01:29, 25 December 2021 (UTC)

See meta:Interwiki map for much more. The one that once surprised me the most was google:something . They are normally for wikis or at least information sites. PrimeHunter (talk) 03:24, 25 December 2021 (UTC)

Question about the Hadwiger–Nelson problem

I'm having a hard time understanding something about the Hadwiger–Nelson problem. In the introduction, it says "The answer is unknown, but has been narrowed down to one of the numbers 5, 6 or 7. The correct value may depend on the choice of axioms for set theory."

I don't have access to the reference and I don't understand how this could be. Suppose one set of axioms has a theorem that the minimum number is 5 and another set of axioms has a theorem that the minimum number is 6. It seems to me that either there is an example for which the solution is 5 or there is not. If there is such an example then something is wrong with the second set of axioms; if there is not such an example, then there is something wrong with the first set of axioms.

Can someone explain this? Bubba73 ^{You talkin' to me?} 23:56, 22 December 2021 (UTC)

You're approaching this from a very Platonist mindset, with your statement that either there is an example or there isn't. From a Platonist perspective, yes, there's something wrong with one of the sets of axioms: it doesn't correctly describe the universe. That doesn't make it not a choice of axioms for set theory.--108.36.85.111 (talk) 00:48, 23 December 2021 (UTC)

I know there are different sets of axioms, but one of them would not be applicable to this problem, since it gives the wrong answer. I remember seeing a video of a talk by Raymond Smullyan in which he said something like "with this set of axioms, the bridge is sturdy and with this other set of axioms the bridge will fall down", and implied that it can't be both ways. Bubba73 ^{You talkin' to me?} 01:14, 23 December 2021 (UTC)

Well, thinking about it some more, maybe whether or not you can prove that the proposed example is or is not the minimum depends on the set of axioms. Is it something like that? Bubba73 ^{You talkin' to me?} 01:32, 23 December 2021 (UTC)

The points in the plane correspond to the set of pairs of two real numbers. I suspect the issue is that the properties of the set of real numbers have a surprising dependence on the axioms of the used set theory. The continuum hypothesis is probably the most famous example. "The set of real numbers" simply isn't as clear a thing as most people think, so neither is "the points on a line" or "the points in the plane". But I may be out of my depth. The answer to the Hadwiger–Nelson problem may depend on what we mean by "points". Mathematicians formalize such things by starting with axioms which can be chosen differently. PrimeHunter (talk) 01:47, 23 December 2021 (UTC)

I think Continuum hypothesis could use a less demanding formulation without terms like cardinality and bijection. Something like:

There is no set A which satisfies both:

Any function from integers to A will miss some values in A.

Any function from A to real numbers will miss some real numbers.

It sounds like this has to be either true or false but it's not that simple. I don't know whether it's related to the Hadwiger–Nelson problem. PrimeHunter (talk) 02:09, 23 December 2021 (UTC)

Yes, this is very counterintuitive. However, under some axiomatization of mathematics (especially one that includes the axiom of choice), it may be possible to prove that a coloring of the plane exists with certain properties, without being able to construct a specific example. Additionally, the example that is proven to exist may not be able to be described in finitely many words or symbols. (Note that the set of finitely describable things is countably infinite, so in any set of axioms there must be indescribable colorings.) So if another set of axioms could be used to prove that no such example exists, there would be no way to use the example from the first set to prove it wrong (since proofs must have finite size). Danstronger (talk) 02:22, 23 December 2021 (UTC)

To elaborate on the contribution by 108.36.85.111, the assertion that "either there is an example for which the solution is 5 or there is not" is an instance of the axiom $P\lor \neg P,$ known as the law of excluded middle. An equivalent principle is $\neg \neg P\rightarrow P$: if the possibility that $P$ is false has been excluded, then $P$ must be true. The vast majority of mathematicians accept this without question, and for most of mathematics as she is practised this works just fine. There is, however, a school of thought, known as constructivism, that rejects the universal applicability of the principle, of which Brouwer's intuitionism is an early example. The enthusiastic embrace of the principle is called (mathematical) Platonism. In essence, the issue boils down to this. To do mathematics, you have to play by the rules. Whose rules? Do these rules "objectively" exist because they are valid in the one-and-only mathematical universe and we just need to discover them, or do these rules spring from the human mind, so that different choices for the rules of the game create different mathematical universes, each of which – if it does not collapse by being inconsistent – is as valid as any other?  --Lambiam 08:48, 23 December 2021 (UTC)

Thank you for those comments. I am definitely thinking that it is or it isn't.

The Continuum Hypothesis may or may not be true. There may be a set with cardinality in between the integers and the reals. But if you proposes such a set, the ZFC axioms aren't strong enough to prove it one way or the other. OK, no problem (with me).

In this problem, it was known for a long time that 7 colors are sufficient and there were examples showing that at least 4 colors are necessary. Then 3 years ago someone came up with an example that requires 5 colors. It took a lot of computer work to prove it.

Suppose someone proposed an example that may require 6 colors. I expect that it would be so large that it would be way beyond what we can check on computers. But it should be computable, in the Turing sense.

But in this case, the statement is that you could have one set of axioms that proves that 6 colors are required and another set of axioms that prove that it can be done with 5 colors. This is a quite different situation than with the Contimuum Hypothesis. I am still not comfortable with that. Bubba73 ^{You talkin' to me?} 20:13, 24 December 2021 (UTC)

You're focusing on finite obstructions, while the question is about colorings of the entire plane. There's a result that the question can be reduced to the finite questions, but that result requires the axiom of choice. Without choice, it's conceivable that every finite unit distance graph can be 5 colored, but the full unit distance graph can only be 6 colored.

Also, finite can be more complicated than you realize, because of nonstandard models. It may be that every true finite unit-distance graph can be 5 colored, but there's a nonstandard model with nonstandard "finite" graphs, one of which requires 6 colors.

For that second point, by way of analogy, consider that (by Gödel) ZFC cannot settle whether or not it proves 0=1. But a proof is a finite object which can be verified, so if someone shows you a proof of 0=1 from ZFC, you'll be able to confirm it. This is analogous to the situation with finite graphs and their chromatic numbers. What's happening is that there is no true proof of 0=1 from ZFC, but there are models with nonstandard pseudo-proofs.108.36.85.111 (talk) 20:58, 24 December 2021 (UTC)

You are right - I was thinking of the finite examples. If it is infinite, then my reasoning is out the window. Bubba73 ^{You talkin' to me?} 20:18, 26 December 2021 (UTC)

So first of all, I agree with the Platonist approach to this problem. Saying that the answer "may depend on the axioms" is misleading. The von Neumann universe is well-specified up to a canonical isomorphism; any statement in the language of set theory is either true or false. It doesn't "depend on the axioms"; rather, what may be the case is that we don't know which are the correct axioms to use.

That said, I think one of the conceptual difficulties you're running into here is the notion of what it means to have an "example" requiring (say) seven colors. There is no need for an "example" to be definable, or at least it's not obvious to me that there is. The article says [b]y the de Bruijn–Erdős theorem, a result of de Bruijn & Erdős (1951), the problem is equivalent (under the assumption of the axiom of choice) to that of finding the largest possible chromatic number of a finite unit distance graph. But I gather that "unit distance" means as embedded in the plane, and that embedding doesn't have to be something you have a name for. Yes, once you have the graph, finding its chromatic number is a finite problem, but it's not obvious that the question of whether a given graph can be embedded in the plane in such a way that all edges have length 1 is a finite problem.

Then again, maybe it is. I haven't thought about it deeply. It does kind of feel as though you might be able to rework it as a question about intersections of circular arcs or some such thing. If that's the case, then I think it would be pretty strange if ZFC didn't decide the answer, because it would be a $\Pi _{1}^{0}$ question, and while there are $\Pi _{1}^{0}$ statements that ZFC doesn't decide, those statements are all true, and generally just reflect higher consistency strength than ZFC. It doesn't feel like a problem whose answer would depend on large cardinals. If it were, that would be really cool, though. --Trovatore (talk) 22:14, 24 December 2021 (UTC)

Whether a finite graph has a unit distance presentation should be decidable by Tarski's decidability of the theory of real closed fields. If E is the edge relation of your graph (with vertices $v_{0},\dots ,v_{n-1}$), the graph being unit distance presentable can be expressed as$\exists x_{0},y_{0},\dots ,x_{n-1},y_{n-1}\bigwedge _{(v_{i},v_{j})\in E}(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}=1\wedge \bigwedge _{(v_{i},v_{j})\not \in E}(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}\neq 1$.108.36.85.111 (talk) 23:16, 24 December 2021 (UTC)

Note that there are well-known examples where something fairly concrete-sounding depends on the axiom of choice. See Banach-Tarski paradox and non-measurable set. Danstronger (talk) 15:57, 25 December 2021 (UTC)

See also the axiom of determinacy, which (I suspect) appears intuitively plausible to Platonists, but is incompatible with AC. So if the magical Banach-Tarski decomposition is possible in your mathematical universe, there exists a set $X$ of infinite sequences of natural numbers such that neither player in the game on $X$ has a winning strategy.  --Lambiam 23:10, 25 December 2021 (UTC)

There is admittedly an intuition in favor of AD, but the one for AC is much more direct. So for Platonists, pretty much inevitably, AC is going to win.

Note by the way that full general determinacy; that is, the intuition that any game of perfect information should admit a winning strategy for one of the players, actually implies AC (and is therefore immediately inconsistent). To see this, let X be a collection of nonempty sets, and play a two-move game where the first player chooses an element x of X, and the second player then chooses an element of x. Clearly the first player cannot have a winning strategy. However, a winning strategy for the second player immediately gives you a choice function for X. --Trovatore (talk) 08:12, 26 December 2021 (UTC)

• Thanks for the input. everyone. This has gotten deeper than I can understand. Bubba73 ^{You talkin' to me?} 20:22, 26 December 2021 (UTC)