From what I can tell, all of mathematics has geometric representations. Euclidean or otherwise. Are there any mathematical objects which do not have this quality?
>>7975001
>geometric representations
what is this ?
>>7975012
My excuse if that is not a precise term. But what I mean is, for example, you can visualize the real number line by a graphical numberline. I specifically did not use the word visualize because even with things like 4D geometry, you can't truly visualize it, but it certainly has a geometric interpretation.
>>7975030
This is so vague it's unfalsifiable and therefor not really worth talking about.
>>7975001
anything that occurs in an infinite-dimensional space is probably not "geometrically representable" as you mean it
>>7975041
My question is vague? Or the idea of a geometric interpretation?
If you are talking about the latter than I must assume you are very casual with your mathematics.
>>7975050
But your example can be represented geometrically.
>>7975050
We can define a notion of curvature on Banach Manifolds.
>>7975001
>all of mathematics has geometric representations
>all of mathematics has representations
Now define geometric.
>>7975001
>mathematics has geometric representations
NO SHIT SHERLOCK
maybe Geometry is a basis for Mathematics, ya think?!
>>7975428
Wow what a terrible sequence of words tbhfam
>>7975428
Now read the rest of the post.
>>7975428
geometry has nothing to do with mathematics
represent a group
>>7976191
[math] \rho :G \to GL\left( V \right)[/math]
>>7975001
Key insight to your question comes from Godel's Incompleteness Theorem, which states that, given any system that is logical and computable (can be used to come to conclusions), there will be statements about objects in that system that are not provable: all mathematical systems are limited in some regard.
This insight gives us some substantial measure by which to measure some objective notion of 'power' of a mathematical system. It follows that we seek to develop new forms of mathematics that are strictly more powerful in some regard than systems in antiquity. The question you're asking becomes, then, whether or not it's possible that there is some mathematical object which, under the axioms of its contextual mathematical system, provides some measure of conclusive 'power' that is completely disjoint from that systems encompassing geometric objects. In other words, is there a non-trivial mathematical object that neither subsumes nor is subsumed by the body of all geometric objects under all possible axiomized forms of mathematics?
Observe the nature of this question. Over all axiomized forms of mathematics, and all possible objects in those systems (a greater infinity), is there one which is disjoint from the group of all geometric objects? We know that if your question can be answered, there exists a Turing machine that will compute the answer (by the Church-Turing thesis). Note, however, that Turing machines have the property that they can be enumerated countably, whereas the equivalent input to such a function contains objects (for example, the object represented by 'all possible problems') that are not able to be represented in a countably infinite way, so representing a greater infinity. By means of the pidgeonhole principle, your question can be shown to not have a computably verifiable answer.
(a+b)^2 has not geometric representation.
>>7976325
of course it has
>>7976325
"The area of a square having side length (a+b)."
>>7975030
Some mathematical objects really can't be visualized, like Suslin line. It's just too bizarre.
>>7976299
This maybe the only read-worthy answer in this thread
just when you think your theory is safe from the geometrytards, some fucking algebraic topologist comes along and computes homology groups of your structures, for absolutely no fucking reason, and then an entire field of study is based off of that guys 'work', and he wins a fields medal, and everyone forgets about your work
next thing you know, this structure you described to represent an abstract number theoretical concept with no relation to geometry WHATSOEVER is being mutated and eventually it's remembered as some perversion of the original concept with n-dimensional holes for n>27 in some weird n+1 dimensional space
I MEAN IT'S FUCKING NUMBER THEORY, THAT THING WAS INVENTED JUST TO MONITOR IDEALS IN SPECIFIC INTEGRAL RINGS
I FUCKING HATE TOPOLOGISTS
>>7976608
>I MEAN IT'S FUCKING NUMBER THEORY, THAT THING WAS INVENTED JUST TO MONITOR IDEALS IN SPECIFIC INTEGRAL RINGS
If describing the integral rings geometrically helps people monitor ideals, then I don't see what the problem is.
>>7976608
lmao I love topology and I hope this is gonna be my life
>>7976391
I am making a distinction between things which can be visualized, and geometric objects in general.
>>7976145
Not so autistic as you may think. OP didn't post with enough clarity for us to have any confidence that he understood the common notion of geometry that the rest of us use. Often times the mathematically inclined will invent notions from intuition, use them for awhile, and then forget that they don't have a formal notion of what they mean outside their head.
Like me for example!
>>7976608
>for absolutely no fucking reason
If I had a chair to fall off of, this would be the when I did.
Not sure if I should offer you a tissue of a sponge...
>>7976974
>a tissue or a sponge
>that typo
Someone on /sci/ asked this question and it got no replies. I've been thinking about it off and on for a few days and haven't cracked it yet:
Can a circle be divided by chords into equal area parts without any of those parts being congruent?
I have proven this can't be done with less than 6 equal parts but see no way to prove it can't be done at all.
>>7976608
i have never laughed so hard at a /sci/ post
>>7977023
If it can be done with three chords then the answer is yes, it can be done. If you're trying to prove it can't be done at all then something is amiss in the way you're looking at the problem. Can you link it? Do you have access to a /sci/ archive?
>>7976608
Top zoz verified post
>>7977159
It can't be done with three chords, that's easy to prove.
>>7977202
May I see the proof?
>>7977205
I would have to draw a bunch of figures in paint, figure it out yourself.
>>7977209
>that's easy to prove
>go figure it out yourself
I would say something else, but this sequence of posts probably says enough on its own.
>>7977213
Holy shit you're dumb. Take a piece of paper and draw 4 circles. Then then think about how there are essentially only three ways to split the circle up using 3 chords. Then realize that 3 of them must have congruent pieces and the last cannot have an equal distribution of area. I'm not going to spend time on paint drawing it out for you when you can do it yourself on paper.
>>7977217
*only 4 ways to split the circle up
>>7977202
So you can't have chords intersect or what?
>>7977240
Huh? Of course they can intersect. Where did I say they can't?
And here's your shitty paint pic, condescending fuck.
>>7977241
Middle figure, two intersections. It's possible to make each of the six compartments have the same area. I now officially have no idea which problem you're trying to solve.
>>7977248
Jesus. Read the fucking problem again. Or just leave since you clearly don't have the mental capability to help solve this.
Can a circle be divided by chords into equal area parts without any of those parts being congruent?
>>7977248
All of these divisions have equal area compartments (except the 4th), that's the fucking point. The question is can you have equal area compartments *without any being congruent*.
>>7977251
Either you're using some non-standard definition of 'congruent' or you're trolling or: It's possible to do it with three chords and two intersections.
>>7977217
Regardless of whether or not I knew the solution you gave before I asked the question, imagine if you had begun an academic paper where you introduce a concept seemingly easy to you with these words.
You're not ready.
>>7977265
I'm using the standard definition of geometrical congruence. How is this hard to understand?
https://en.wikipedia.org/wiki/Congruence_(geometry)
>>7977241
This is far from a formal proof, anon.
>>7977265
If you cut a circle with three chords in which two of them intersect and every piece has equal area, then you have 4 congruent pieces on the sides and 2 congruent pieces in the middle. This is the easiest one to see.
>>7977273
If I was writing an academic paper I would have the proof written down rigorously. How the fuck is this conversation analogous? You implied that I did not have a proof simply because I did not want to sketch it out for you, and then you showed that you didn't even understand the (simple) question being asked.
>>7977276
Never said it was.
>>7977280
This is analogous because, when you write a paper, people will ask you questions that you may deem to be dumb. If you condescend and throw insults, you're much less likely to be published.
I implied that you had not given the proof which you claimed you had. No need to become defensive.
>>7977283
>This is analogous because, when you write a paper
*bzzzzzzt* wrong. I'm not writing a paper, nor is a conversation on 4chan about a math problem analagous to writing a paper, so just stop. If you have nothing mathematical to add and are just going to whine about how I treated you, then fuck off.
>>7977274
Without. The question asks for incongruent areas.
>>7977292
And...?
>>7977292
shut the fuck up already fucking shitposter
>>7977280
Hi separate anon (that's probably manipulated the other anon that's been working on this problem). I want to know how and why you think etc.
>>7977292
incongruent
not congruent
not satisfying all properties of congruence
...
>>7977302
Are you one of those schizophrenic namefags or are you just deliberately trying to waste my time?
>>7977310
It can be done with three chords. If you disagree then either show your work or stop trolling.
>>7977331
No it can't. Post a picture of a circle split by three chords with all parts having equal area and none being congruent. This is impossible and you won't do it because you are a shithead.
>>7977331
a picture was already posted with a proof you fucking schizophrenic namefag
you idiots are all the same, are you all fucking 12?
>>7977341
I don't have the means to post a picture at the moment. I also don't care about the question itself, I care about the events behind the question as well as the number of anons working on it and why they're working on it and why they haven't solved it yet. I'm not going to discuss trivial geometry here.
>>7977366
>trivial geometry
>your insight on it was "If it can be done with three chords then the answer is yes, it can be done." when it clearly can't be done with three chords
fuck off you retarded faggot
>>7977370
Literally "if it can be done then it can be done"
I had to slap myself after I read this.
>>7977366
Then fuck off.
>>7977292
Fucking retarded tripfag get the fuck off my board
f: C->C
>>7977613
I already posted the proof you can't. Do it or shut the fuck up.
>>7977023
I have made some progress on the proof. Ignore the schizophrenic namefag shitposting the thread up.
At first I thought that the first step would be to prove that when a circle is divided into equal area parts by chords, at least two of the chords must be congruent. I started by observing that a chord's length is uniquely determined by the number of equal area parts it divides the circle into. If the circle is divided into P parts then a chord divides it into n and P-n parts with n =< P-n. So we can identify which chords are unique and which are congruent by labeling them with the number n. If and only if two chords have the same n are they congruent. If we look at the chord which most evenly divides the circle, then the sum of the numbers of chords and intersections within the part of the circle it divides into n parts must be n-1. I then thought I could pigeonhole it to show that this number of chords and intersections would restrict the number of unique chords in the circle t be less than the number of chords.
However I soon realized that it is possible to at least sketch an example with 5 chords uniquely characterized by n's 1 through 5. It is not clear if this would actually divide a circle equally in practice but this seems to invalidate my thinking about the pigeonhole. I thought that proving there must be congruent chords would allow me to prove there must be congruent shapes, but now I'm not so sure.
