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Geometric Representation of Numbers

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Sunshine7
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13 Nov 2011, 3:56 pm

Some background: my friends love to confound me with nonsensical questions, because they know I'm the type of person who cannot let a question go, even if it has no answer, and when I arrive at one, it will be so rigidly logical that no matter how ridiculous it sounds it must be correct. My previous stumper was: if I ate myself, would my mass be multiplied by 2, zero, or remain the same (no, conservation law is not necessarily the answer, see Noether's theorem)?

The latest one is: Square + cube = ?

My latest venture into this question let me into exploration of hypercubes and graph theory http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagram
however, the geometric representation of numbers does not seem to be a vast field in itself. Does anybody with a math degree know anything about this, in particular: http://en.wikipedia.org/wiki/Minkowski_space



ruveyn
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13 Nov 2011, 5:31 pm

Sunshine7 wrote:
Some background: my friends love to confound me with nonsensical questions, because they know I'm the type of person who cannot let a question go, even if it has no answer, and when I arrive at one, it will be so rigidly logical that no matter how ridiculous it sounds it must be correct. My previous stumper was: if I ate myself, would my mass be multiplied by 2, zero, or remain the same (no, conservation law is not necessarily the answer, see Noether's theorem)?

The latest one is: Square + cube = ?

My latest venture into this question let me into exploration of hypercubes and graph theory http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagram
however, the geometric representation of numbers does not seem to be a vast field in itself. Does anybody with a math degree know anything about this, in particular: http://en.wikipedia.org/wiki/Minkowski_space


This is the first time I have seen these diagrams. Way cool! Many thanks for bringing it up!

ruveyn



JSNS
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13 Nov 2011, 7:17 pm

Quote:
The latest one is: Square + cube = ?

Define "+" in this equation.



cw10
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14 Nov 2011, 5:51 am

Image

Something like this I would imagine.



cw10
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14 Nov 2011, 5:54 am

http://en.wikipedia.org/wiki/Menger_sponge



Sunshine7
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14 Nov 2011, 6:35 am

Quote:
Define "+" in this equation.


In the spirit of the original question, delivered when everybody was not just a little inebriated, you can define it any way you want.

As for the Menger sponge: interesting. but it's more like cube + smaller cube + even smaller cube ad infinitum...all strictly in 3 dimensions. Although I haven't studied fractals formally, so I dont know how they investigate dimensionality.



cw10
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14 Nov 2011, 7:09 am

Sunshine7 wrote:
Quote:
Define "+" in this equation.


In the spirit of the original question, delivered when everybody was not just a little inebriated, you can define it any way you want.

As for the Menger sponge: interesting. but it's more like cube + smaller cube + even smaller cube ad infinitum...all strictly in 3 dimensions. Although I haven't studied fractals formally, so I dont know how they investigate dimensionality.


Fractals aren't 2 or 3 dimensional, they can something in between when dealing with a "3 dimensional" fractal.

3+2=5
5/2=2.5



ruveyn
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14 Nov 2011, 8:35 am

cw10 wrote:
Image

Something like this I would imagine.


That is more like subtraction than addition. The Koch triangle does add stuff to get a triangle with fractile boundaries.

ruveyn



mar00
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14 Nov 2011, 9:19 am

Image
Now just imagine this guy having one square ear.



JSNS
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14 Nov 2011, 10:54 am

I think; You would have the mass of the "mandelbrunch" whose insides are 1-1 with its outsides ; for your question.



WoodenBoy
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14 Nov 2011, 11:06 am

"Square + cube" is actually quite meaningful for me.

There is a thing in geometry called the "Minkowski sum", which I don't think has anything in common with the Minkowski space stuff in relativity.

To obtain the Minkowski sum of two objects, you essentially replace each point in one object with a copy of the other (you get the same result whichever way round you do it).

The most relevant picture I can find is here:

Image

You can think of that 3d figure as the Minkowski sum of a line segment and a cube.



ruveyn
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14 Nov 2011, 11:16 am

WoodenBoy wrote:

You can think of that 3d figure as the Minkowski sum of a line segment and a cube.


More like a Cartesian product than a sum. A donut is the Cartesian product of two circles on planes perpendicular to each other. A cylinder is the Cartesian product of a circle and a line perpendicular to the center of the circle.

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cw10
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14 Nov 2011, 11:55 am

WoodenBoy wrote:
"Square + cube" is actually quite meaningful for me.

There is a thing in geometry called the "Minkowski sum", which I don't think has anything in common with the Minkowski space stuff in relativity.

To obtain the Minkowski sum of two objects, you essentially replace each point in one object with a copy of the other (you get the same result whichever way round you do it).

The most relevant picture I can find is here:

Image

You can think of that 3d figure as the Minkowski sum of a line segment and a cube.


Now THAT is interesting.



cw10
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14 Nov 2011, 12:00 pm

cw10 wrote:
WoodenBoy wrote:
"Square + cube" is actually quite meaningful for me.

There is a thing in geometry called the "Minkowski sum", which I don't think has anything in common with the Minkowski space stuff in relativity.

To obtain the Minkowski sum of two objects, you essentially replace each point in one object with a copy of the other (you get the same result whichever way round you do it).

The most relevant picture I can find is here:

Image

You can think of that 3d figure as the Minkowski sum of a line segment and a cube.


Now THAT is interesting.


Now here's another challenge. Find a mineral that resembles that structure.



graywyvern
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14 Nov 2011, 12:05 pm

i thought this was going to be about stuff like this:

http://mathworld.wolfram.com/PentagonalNumber.html


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Sunshine7
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15 Nov 2011, 5:46 am

Ah yes, I've looked at the Minkowsky sum. Somebody mentioned pentagonal numbers: I've looked at the more general case of polyhedral numbers (Any X^y - or depending on the programming language you're used to, X**y - where x and y are natural numbers). I've wondered if Minkowsky addition can yield anything if you add 2 polyhedral numbers, although so Minkowsky addition has the drawback on being defined on the real, not natural, numbers.