Platonic solids in Non-Euclidean space

[imatfaal]

 

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There is no mechanism with in the neuro-chemistry of the brain which can account for how the mathematicians can access the ideal world of the platonic realm and obtain absolute mathematical truths. However we have a mechanism for how mathematicians access the ideal world of numbers in the platonic realm. ...

 

 

 

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The chemistry behind mathematicians' brains is the same than that behind the rest of us. They are affected by the same drugs, chemicals... Moreover there is not "absolute mathematical truths" in our modern understanding of maths.

 

On reading this exchange (part of a long thread on definition of religion) - something occurred to me. Do the rules governing regular 3-d solids change in non-euclidean space?

 

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

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

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

 

etc...

[ydoaPs]

There is no mechanism with in the neuro-chemistry of the brain which can account for how the mathematicians can access the ideal world of the platonic realm and obtain absolute mathematical truths. However we have a mechanism for how mathematicians access the ideal world of numbers in the platonic realm.

That's probably because there is no Platonic Realm. Maths are invented conceptual constructs which are (sometimes) descriptive approximations of reality.

[DevilSolution]

Where does non-euclidean space exist?? i dont get it

[imatfaal]

Where does non-euclidean space exist?? i dont get it

 

It's maths - why does it have to exist?

[DevilSolution]

It's maths - why does it have to exist?

 

I still don't understand.

 

So let me re-phrase; Where you say "Do the rules governing regular 3-d solids change in non-euclidean space?" you must first have a base premise for exactly what this "non-euclidean space" is. You can say euclidean space is a mathematical concept which therefor might not exist, but you've still offered nothing to for us to base the rules of 3-d solids on.

 

so to answer your question as is; No, the rules of 3-d objects do not change outside euclidean space, as there is nothing outside of euclidean space.

 

To go a little further, if we say the first dimension starts with 2 points (as seen physically with a line), the second is 3 points where the lines connecting are inverse, and the third is created using 4 points, where the third line again is inverse to both the other 2 lines. At this point, with 4 points and 3 lines, there is nothing beyond this that can be conceptualize by us, it can be hypothesized but never physically imagined. Also mathematically how would you attempt to explain shapes in a non euclidean or cartesian way?

Edited March 1, 2013 by DevilSolution

[imatfaal]

I still don't understand.

 

So let me re-phrase; Where you say "Do the rules governing regular 3-d solids change in non-euclidean space?" you must first have a base premise for exactly what this "non-euclidean space" is. You can say euclidean space is a mathematical concept which therefor might not exist, but you've still offered nothing to for us to base the rules of 3-d solids on.

Non-euclidean space is space in which parallel line converge or diverge - ie the space is proof that Euclid's parallel postulate is not necessarily true. The other way of thinking of it is that triangle have other than 180 degrees in total internal energy angles.

 

 

so to answer your question as is; No, the rules of 3-d objects do not change outside euclidean space, as there is nothing outside of euclidean space.

Hyperbolic geometry is very well established - it's just I am not an expert.

 

 

To go a little further, if we say the first dimension starts with 2 points (as seen physically with a line), the second is 3 points where the lines connecting are inverse, and the third is created using 4 points, where the third line again is inverse to both the other 2 lines. At this point, with 4 points and 3 lines, there is nothing beyond this that can be conceptualize by us, it can be hypothesized but never physically imagined. Also mathematically how would you attempt to explain shapes in a non euclidean or cartesian way?

 

I have solved a four-dimensional rubik's cube on line (I will see if I can dig out the link) so I disagree that beyond 3 dimensions is impossible to conceptualize. And other geometries are easily shown through analogy to two dimensional curved surfaces.

Edited March 4, 2013 by imatfaal
for very weird typo

[ydoaPs]

The other way of thinking of it is that triangle have other than 180 degrees in total internal energy.

We can make a pyramid with triangles. This means, in a noneuclidean space, we can make noneuclidean triangles. And distances are defined by the pythagorean theorem (well, a generalized version): s²=x²+y²+z². So, let's take a sphere-the set of all points in the space equidistant from a defined point). Now do you think the 'rules' (I'm not exactly sure what that means) change with the geometry of the space?

[DevilSolution]

I have solved a four-dimensional rubik's cube on line (I will see if I can dig out the link) so I disagree that beyond 3 dimensions is impossible to conceptualize. And other geometries are easily shown through analogy to two dimensional curved surfaces.

 

I presume this rubiks cube is a 3 dimensional representation of 4 dimensions and that you didnt physically figure it out; I.E. you don't physically have a 4 dimensional cube.

 

Let me offer you this; Remove your conceptualized 4th dimension and now try imagining a spherical object that can be wrapped around itself in any of its euclidean axis's. PERSONALLY, i find this impossible, but feel free to explain how you conceptualize it.

 

What i think this means is that where 2 dimensional geometry can be used to explain 3 dimensional shapes, we cant do the same for 4 dimensional shapes I.E. where other self consistent geometries can.

 

I dont know enough of other geometries that can consistently define 2-3-4-d shapes to offer any explanation on the nature of thats shapes existence outside of the euclidean geometry (which is the only one which i can currently comprehend).

Edited March 4, 2013 by DevilSolution

[imatfaal]

We can make a pyramid with triangles. This means, in a noneuclidean space, we can make noneuclidean triangles. And distances are defined by the pythagorean theorem (well, a generalized version): s²=x²+y²+z². So, let's take a sphere-the set of all points in the space equidistant from a defined point). Now do you think the 'rules' (I'm not exactly sure what that means) change with the geometry of the space?

 

 

"internal energy of a triangle" that was my fingers doing the typing rather than my brain.

 

By rules I was thinking of the sort of thing that says that there are only 5 regular convex shapes creatable by the regular polygons. But if the angles no longer need to be constrained could we - for example create a regular polyhedron from the hyperbolic version of hexagons (which in flat space simply tile the plane? Some rules clearly will stay the same - ie the rules relating vertices, sides, and faces, but others seem to depend on the flatness of space.

 

 

I presume this rubiks cube is a 3 dimensional representation of 4 dimensions and that you didnt physically figure it out; I.E. you don't physically have a 4 dimensional cube.

Yes that's why the example shows it is possible to conceptualize the 4 dimension - to actualize it would be another thing entire

ly.

[DevilSolution]

Yes that's why the example shows it is possible to conceptualize the 4 dimension - to actualize it would be another thing entire

ly.

 

This is why im confused. You can suggest taking taking 3-d space and making it flat so were looking at it 2-d; then take the 2-d image and fold it over itself connecting the edges etc etc. We could build a bigger and bigger picture using analogies and examples but to physically imagine anything bigger than the 3rd i find is impossible.

[imatfaal]

This is why im confused. You can suggest taking taking 3-d space and making it flat so were looking at it 2-d; then take the 2-d image and fold it over itself connecting the edges etc etc. We could build a bigger and bigger picture using analogies and examples but to physically imagine anything bigger than the 3rd i find is impossible.

 

"physically imagine" - this is where I am confused. I can imagine - and I can make physical; they are completely different. Any thought of making 4d physically, actually, is incorrect. But if we have clever analogies and aids to thought, good diagramming and representation, and possible a little mind expanding pharmacology - we can imagine (perhaps completely wrongly) or more honestly begin to imagine 4d objects.

 

My analogy of 2d curved space is the existence of 2d curved surfaces (ie the sphere) which exist in our 3d world - it is not making a 3d curved surface flat, it is looking at a 2d curved space and trying to conceptualise how that works and upgrading to 3 dimensions of curved space in a 4d world (this is beyond me)

[DevilSolution]

I'm basing the word "imagine" on "image" in that you can only imagine something that physically exists. In the same way we might say a blind person cant "imagine" the colour blue, a human cant see an image in any higher dimension than the 3rd.

 

As previously stated i dont have any understanding of any geometric system other than euclidean so i cant imagine how a shape would act outside of that system. I still suppose you would need to offer which specific geometric system you mean.

[DevilSolution]

Ive done some reading into the geometries involved, non-euclid geometry isnt 2-d i dont believe, if you take a 3-d sphere and call any point on it the origin A; then draw two lines at 90 degree on A called B and C, like slicing 1/4 of an apple. Now of this 1/4 if we draw 2 new lines across B and C that are parallel (same angle) but not equal in distance called D and E they will both have 2 90 degree angles from the other 2 lines, the problem with euclid's 5th postulate is that these 2 parallel lines eventually meet at some point beyond our current triangle's (i will make an illustration if my explanation is confusing). This essentially means that on a curved surface (3-d), the axioms (or 5th postulate) of 2-d geometry do not hold, however im slightly confused with the 5th postulate anyway considering it is only intuitive and obvious and not actually based on the other 4 postulates.

 

All in all i think 3-d space and the 2-d cortesian plane do not correlate relative to circles, all other the other geometry seems sound. Circles are strange by nature when we try to represent them in numbers, for example in computer graphics, to define the points of circle we multiply the radius by 360 and draw that many vertices, this generally (with the exception of resolution) will always give each vertex of the circle a separate pixel; In the reality however there is no definite points of a circle, its infinite. Unlike euclid triangles which work perfectly on a cortesian plane to represent a flat surface and thats why computer graphics use polygons and not circle geometry.

 

Non-euclidean geometry has real scientific purpose due to things like sailing and air travel where we are traveling on a spherical object instead of a 2-d plane. 2-d geometry also has its scientific purpose when 1 dimension isnt needed, such that we are calculating height of a building based on its angle and distance, the only data needed is forward to back and up to down, the side to side isnt relative.

 

To answer ">Do the rules governing regular 3-d solids change in non-euclidean space?", based on what ive read, yes but only the 5th postulate is contradicted so the rest of the "laws" would stay the same......as to what this means in reality other that whats been stated im not really sure. I presume anything circle related is totally different, all other 3-d shapes act in accordance to the 4 other euclidean axioms and DONT change in non-euclidean geometries.

 

">and possible a little mind expanding pharmacology".......touche

Edited March 7, 2013 by DevilSolution