A challenge

Higher-dimensional geometry (previously "Polyshapes").

A challenge

Postby Twix18 » Sun May 08, 2005 6:41 am

make a negitive 3-D shape.
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Postby jinydu » Sun May 08, 2005 7:16 am

As far as we know, it can't be done.

Nevertheless, there was a fun poem about a cube with volume -1. Unfortunately, I no longer have the look, but you could probably search for it on Google.
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Postby brasileiro » Sun May 08, 2005 8:04 pm

I haven't thought about this too much... wouldn't that just cause a void?
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Postby jinydu » Sun May 08, 2005 8:09 pm

If you really want to persist with the question: What do you mean by a "negative 3D shape"? Do you mean it has negative volume, or surface area, or what?
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Postby brasileiro » Sun May 08, 2005 8:40 pm

.... yeah... what he said...
"Science without religion is lame. Religion without science is blind."
~Albert Einstein

If you teach someone in the 2nd dimension, they will live in the 2nd dimension... if you teach them in the 3rd, they will live in the 3rd.
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Postby houserichichi » Mon May 09, 2005 12:37 am

Depends what definition/type of dimension you mean...
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Postby wendy » Mon May 09, 2005 4:35 am

The trap here is to think that a cube of -1 inch side would have a surface area of 6*(-1)^2 sq inches, ie 6 square inches. Adding this to another cube of 1 inch side, means the edges and volume disappears leaving just the vertices and edges.

This assumes that the prismatic product continues to map onto the numeric product for negative numbers. This is not the case.

Firstly, a measure of -1 can be read as a break in an elsewise contigious line, and the resulting square and cube are likewise absences in the plane and space. Under this, -1 * -1 = -1, and multiplying negative and positive numbers is not defined. The particular definition is valid in polytope density distributions.

Alternately, we have the matrix determinate proposition, where we can find the determinate of a diagonal matrix. The absolute value of the determainate is the volume of the column-vectors, or row-vectors, and the sign is a parity thing from the underlying pyritotopic group. Exchanging any pair of axies makes the thing change sign.

In any case, while negative polytopes are useful, you must never assume that negative means negative means negative. That is, you are on thin ice, and must exercise care in reading meaning into negative numbers.

The most negative dimension currently observed is -2. This occurs as the low point of the desarge configuration, when rendered as a pondered simplex.
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Postby houserichichi » Mon May 09, 2005 5:28 am

I have to ask, what are both the "prismatic product" and the "pyritotopic group". I'm not familiar with either of them and google turned up nothing.
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Postby wendy » Mon May 09, 2005 11:17 am

The prismatic product is one of the four geometric products. It produces the sequence of polytopes line, square, cube, tesseract, ..., It generalises to any combinations of bases, eg

pentagon * hexagon prism

The other three are pyramid, tegum and comb products.

The pyritohedral symmetry is one based on the symmetry of pyrities.

In mathematical terms, it means for a point (x,y,z), even permutations of positions and all change of sign. It has the orbifold 3 * 2.

It generalises to higher dimensions. The symmetry of the snub 24-choron is the point (w,x,y,z), with even permutations and all change of sign. The group in 4D is order 192, with a subgroup of order 576.

The pyritotopic group generalises this.

The cubic has these subgroups in all dimensions.

C = cubic all permutations, all change of sign
C/2 = semicubic all permutations, even change of sign
C/2 = pyritohedral even permutations, all change of sign
C/2 = cubic-rotational even (permutations + change of sign)
C/4 = semicubic - rot even permutations, even change of sign

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Postby brasileiro » Mon May 09, 2005 7:11 pm

Ahhhhhh... I gotchya. I was a little fuzzy on that
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Postby Marek14 » Thu Jul 21, 2005 12:56 pm

Reminds me of a poem I read somewhere on the web (no longer remembering where):

Minus One
------------------------------------
A carpenter named Charlie Bratticks
Who had a taste for mathematics
One summer Tuesday, just for fun
Made a wooden cube side minus one

Though this to you may seem wrong
He made if minus one foot long
Which meant (I hope your brains aren't frothing)
Its length was one foot less than nothing

Its width the same (you're not asleep?)
And likewise minus one foot deep
Giving, when multiplied (be solemn)
Minus one cubic foot of volume

With sweating brow this cube he sawed
Through areas of solid board
For though each cut had minus length
Minus times minus sapped his strength

A second cube he made, but thus
This time each one foot length was plus
Meaning of course that here one put
For volume, plus one cubic foot

So now he had, just for his sins
Two cubes as like as deviant twins
And feeling one should know the worst
He placed the second in the first

One plus, one minus - there's no doubt
The edges simply cancelled out
So did the volume, nothing gained
Only the surfaces remained

Well may you open wide your eyes
For those were now of double size
On something now, thanks to his skill
Took up no room and measured nil

From solid ebony he'd cut
These bulky objects, but
All that remained was now a thin
Black sharply angled sort of skin

Of twelve square feet - which though not small
Weighed nothing, filled no space at all
It stands there yet on Charlies floor
He can't think what to use it for
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Postby jinydu » Thu Jul 21, 2005 2:59 pm

Marek14 wrote:Reminds me of a poem I read somewhere on the web (no longer remembering where):


A quick Google search:

http://astronomy.swin.edu.au/~pbourke/g ... inus1.html
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