tensors

Rambling


If i take a cube and unfold it in a way that it becomes a planar object then
there is degeneracy in terms of dimension - The planar form for 3D is a
Cross Crucifix) - the forth dimension of time added results in a 3D cross
(This is not the only configuration) the form is a translation of the
original shape.


Question?

What symbol *should* represent 11 dimensions?

Kieran

"Aesop" wrote in message
...
Rambling


If i take a cube and unfold it in a way that it becomes a planar object
then
there is degeneracy in terms of dimension - The planar form for 3D is a
Cross Crucifix) - the forth dimension of time added results in a 3D cross
(This is not the only configuration) the form is a translation of the
original shape.


Question?

What symbol *should* represent 11 dimensions?

Kieran


This is an object known as a 'tesseract' - unfolding results in an object
one dimension lower, so it is a trick we can use to visualise higher
dimensions. It is not possible to visualise an 11 dimensional one, but you
can express it mathematically (via topology). There is some good information
on this in the book 'Hyperspace' by Michio Kaku

p.s. Salvador Dali featured the 3D tesseract you mention in his crucifiction
painting

Aesop wrote:
Rambling

It shows...

If i take a cube and unfold it in a way that it becomes a planar object

Wrong. A cube is a 3-D object. It cannot be "unfolded" into a 2-D
object. You are probably thinking about the *surface* of a cube.

There are many topologically distinct ways that 6 squares can be
arranged so that they can be folded up into the surface of a cube.

then
there is degeneracy in terms of dimension - The planar form for 3D is a
Cross Crucifix)

I don't know which definition of the word 'degeneracy' you are using
here,
http://www.onelook.com/?w=degeneracy&ls=a
but in the mathematical sense
http://mathworld.wolfram.com/Degenerate.html
it does not apply to this situation.

- the forth

"fourth"?

dimension of time added results in a 3D cross

I infer you are referring to the shape depicted he
http://en.wikipedia.org/wiki/Tesseract
And where is the 'time' dimension of your 3-D cross? What does it add
to the figure that was absent before it was added?

(This is not the only configuration) the form is a translation of the
original shape.

"translation?" In which direction or into what language? Again your
misuse of the words leads to miscommunication.

Question?

What symbol *should* represent 11 dimensions?

The usual mathematical convention is a special font capital Roman R
followed by a superscript representing the number of dimensions (in
your case, "11"):
http://mathworld.wolfram.com/Dimension.html

Tom Davidson
Richmond, VA

http://en.wikipedia.org/wiki/Tesseract
http://mathworld.wolfram.com/Tesseract.html
http://mathworld.wolfram.com/Hypercube.html

Tom Davidson
Richmond, VA

"tadchem" wrote in message
oups.com...


Aesop wrote:
Rambling

It shows...

If i take a cube and unfold it in a way that it becomes a planar object

Wrong. A cube is a 3-D object. It cannot be "unfolded" into a 2-D
object. You are probably thinking about the *surface* of a cube.

There are many topologically distinct ways that 6 squares can be
arranged so that they can be folded up into the surface of a cube.

then
there is degeneracy in terms of dimension - The planar form for 3D is a
Cross Crucifix)

I don't know which definition of the word 'degeneracy' you are using
here,
http://www.onelook.com/?w=degeneracy&ls=a
but in the mathematical sense
http://mathworld.wolfram.com/Degenerate.html
it does not apply to this situation.

- the forth

"fourth"?

dimension of time added results in a 3D cross

I infer you are referring to the shape depicted he
http://en.wikipedia.org/wiki/Tesseract
And where is the 'time' dimension of your 3-D cross? What does it add
to the figure that was absent before it was added?

(This is not the only configuration) the form is a translation of the
original shape.

"translation?" In which direction or into what language? Again your
misuse of the words leads to miscommunication.

Question?

What symbol *should* represent 11 dimensions?

The usual mathematical convention is a special font capital Roman R
followed by a superscript representing the number of dimensions (in
your case, "11"):
http://mathworld.wolfram.com/Dimension.html

Tom Davidson
Richmond, VA


Tom drop your red pen for a moment please and consider

Degenerate

Having declined or or deteriorated to a lower physical level (2D
representation of a 3D object)

Translation

Any form that changes to another is 'translated' (Rather than the strict
matematical sense)

I am assuming, perhaps incorrectly, you have the intellect to think in
english as well as mathematics

I was thinking about the surface of the cube (granted - hangs head in shame)

I am familiar with the R term - What I was thinking about was unfolding the
3D cross as I did the Cube and adding another dimension to the resulting 2D
object making it a 5D representation of the original object (using the same
scheme as with the cube - cross)

My question was what would be the resulting shape for 11 dimensions

Kieran

"Anthony Smales" wrote in message
...

"Aesop" wrote in message
...
Rambling


If i take a cube and unfold it in a way that it becomes a planar object
then
there is degeneracy in terms of dimension - The planar form for 3D is a
Cross Crucifix) - the forth dimension of time added results in a 3D
cross
(This is not the only configuration) the form is a translation of the
original shape.


Question?

What symbol *should* represent 11 dimensions?

Kieran


This is an object known as a 'tesseract' - unfolding results in an object
one dimension lower, so it is a trick we can use to visualise higher
dimensions. It is not possible to visualise an 11 dimensional one, but you
can express it mathematically (via topology). There is some good
information
on this in the book 'Hyperspace' by Michio Kaku

p.s. Salvador Dali featured the 3D tesseract you mention in his
crucifiction
painting

Man nailed to a 4 dimensional existence?

Aesop wrote:

snip

Degenerate

Having declined or or deteriorated to a lower physical level (2D
representation of a 3D object)

Translation

Any form that changes to another is 'translated' (Rather than the strict
matematical sense)

I am assuming, perhaps incorrectly, you have the intellect to think in
english as well as mathematics

I am educated in the contextual definitions of words in both fields -
my mother was a proofreader for newspapers and my father was a
physicist, my first undergraduate degree was in mathematics (I am now a
chemist). You yourself established the context as mathematics, and I
would have been presumptive to assume you were changing contexts in
mid-discussion. Now that I have your admission, I can adapt. My wife
does it to me all the time - but then she's just an engineer.

I am familiar with the R term - What I was thinking about was unfolding the
3D cross as I did the Cube and adding another dimension to the resulting 2D
object making it a 5D representation of the original object (using the same
scheme as with the cube - cross)

My question was what would be the resulting shape for 11 dimensions

There are 11 distict *nets* for the 3-D cube (that is what your 'cross'
shape obtained from unfolding the surface of cube onto a 2-D surface is
called):
http://mathworld.wolfram.com/Cube.html
http://mathworld.wolfram.com/Net.html

The tesseract (4-dimensional cube) has 261 "nets"
http://mathworld.wolfram.com/Tesseract.html

The number of nets grows rapidly with the order of the hypercube. The
Online Encyclopedia of Integer Sequences
http://www.research.att.com/cgi-bin/...i?Anum=A091159
does not give the number of nets for hypercubes beyond the tesseract,
although they are calculable - not that I know how off the top of my
head.

The page
http://mathworld.wolfram.com/Hypercube.html
displays 2-dimensional projections of the hypercubes of orders
(dimensionality) 2 through 7.

Extrapolating from a study of these we can see the following:

The 11-D hypercube will have 2^11 (2048) vertices (corners), with 11
edges meeting at each vertex from orthogonal (perpendicular) directions
(total 11,264 edges), each edge shared by 10 square faces (total 28,160
square faces), and it gets complicated after that...

HTH

Tom Davidson
Richmond, VA

"Aesop" wrote in message
...

"Anthony Smales" wrote in message
...

"Aesop" wrote in message
...
Rambling


If i take a cube and unfold it in a way that it becomes a planar object
then
there is degeneracy in terms of dimension - The planar form for 3D is a
Cross Crucifix) - the forth dimension of time added results in a 3D
cross
(This is not the only configuration) the form is a translation of the
original shape.


Question?

What symbol *should* represent 11 dimensions?

Kieran


This is an object known as a 'tesseract' - unfolding results in an object
one dimension lower, so it is a trick we can use to visualise higher
dimensions. It is not possible to visualise an 11 dimensional one, but
you
can express it mathematically (via topology). There is some good
information
on this in the book 'Hyperspace' by Michio Kaku

p.s. Salvador Dali featured the 3D tesseract you mention in his
crucifiction
painting

Man nailed to a 4 dimensional existence?

I think its called 'Christus Hypercubus'
here it is: http://mapage.noos.fr/dardelf2/museum5/Dali-corpus.jpg

"tadchem" wrote in message
oups.com...


Aesop wrote:

snip

Degenerate

Having declined or or deteriorated to a lower physical level (2D
representation of a 3D object)

Translation

Any form that changes to another is 'translated' (Rather than the strict
matematical sense)

I am assuming, perhaps incorrectly, you have the intellect to think in
english as well as mathematics

I am educated in the contextual definitions of words in both fields -
my mother was a proofreader for newspapers and my father was a
physicist, my first undergraduate degree was in mathematics (I am now a
chemist). You yourself established the context as mathematics, and I
would have been presumptive to assume you were changing contexts in
mid-discussion. Now that I have your admission, I can adapt. My wife
does it to me all the time - but then she's just an engineer.

I am familiar with the R term - What I was thinking about was unfolding
the
3D cross as I did the Cube and adding another dimension to the resulting
2D
object making it a 5D representation of the original object (using the
same
scheme as with the cube - cross)

My question was what would be the resulting shape for 11 dimensions

There are 11 distict *nets* for the 3-D cube (that is what your 'cross'
shape obtained from unfolding the surface of cube onto a 2-D surface is
called):
http://mathworld.wolfram.com/Cube.html
http://mathworld.wolfram.com/Net.html

The tesseract (4-dimensional cube) has 261 "nets"
http://mathworld.wolfram.com/Tesseract.html

The number of nets grows rapidly with the order of the hypercube. The
Online Encyclopedia of Integer Sequences
http://www.research.att.com/cgi-bin/...i?Anum=A091159
does not give the number of nets for hypercubes beyond the tesseract,
although they are calculable - not that I know how off the top of my
head.

The page
http://mathworld.wolfram.com/Hypercube.html
displays 2-dimensional projections of the hypercubes of orders
(dimensionality) 2 through 7.

Extrapolating from a study of these we can see the following:

The 11-D hypercube will have 2^11 (2048) vertices (corners), with 11
edges meeting at each vertex from orthogonal (perpendicular) directions
(total 11,264 edges), each edge shared by 10 square faces (total 28,160
square faces), and it gets complicated after that...

HTH

Tom Davidson
Richmond, VA

So the Euler number, X would be :
X = V - E + F
X = 2048 - 11264 + 28160 = 18944 (Euler numbers of equivalent surfaces are
equal)

"tadchem" wrote in message
oups.com...


Aesop wrote:

snip

Degenerate

Having declined or or deteriorated to a lower physical level (2D
representation of a 3D object)

Translation

Any form that changes to another is 'translated' (Rather than the strict
matematical sense)

I am assuming, perhaps incorrectly, you have the intellect to think in
english as well as mathematics

I am educated in the contextual definitions of words in both fields -
my mother was a proofreader for newspapers and my father was a
physicist,


Being of - is no guarantee of being

my first undergraduate degree was in mathematics (I am now a
chemist). You yourself established the context as mathematics,

No it was rambling

and I
would have been presumptive to assume you were changing contexts in
mid-discussion. Now that I have your admission, I can adapt. My wife
does it to me all the time - but then


she's just an engineer. (private joke no doubt)



I am familiar with the R term - What I was thinking about was unfolding
the
3D cross as I did the Cube and adding another dimension to the resulting
2D
object making it a 5D representation of the original object (using the
same
scheme as with the cube - cross)

My question was what would be the resulting shape for 11 dimensions

There are 11 distict *nets* for the 3-D cube (that is what your 'cross'
shape obtained from unfolding the surface of cube onto a 2-D surface is
called):
http://mathworld.wolfram.com/Cube.html
http://mathworld.wolfram.com/Net.html

The tesseract (4-dimensional cube) has 261 "nets"
http://mathworld.wolfram.com/Tesseract.html

The number of nets grows rapidly with the order of the hypercube. The
Online Encyclopedia of Integer Sequences

http://www.research.att.com/cgi-bin/...es/eisA.cgi?An
um=A091159
does not give the number of nets for hypercubes beyond the tesseract,
although they are calculable - not that I know how off the top of my
head.

The page
http://mathworld.wolfram.com/Hypercube.html
displays 2-dimensional projections of the hypercubes of orders
(dimensionality) 2 through 7.

Extrapolating from a study of these we can see the following:

The 11-D hypercube will have 2^11 (2048) vertices (corners), with 11
edges meeting at each vertex from orthogonal (perpendicular) directions
(total 11,264 edges), each edge shared by 10 square faces (total 28,160
square faces), and it gets complicated after that...

HTH

Whats HTH?

Tom Davidson
Richmond, VA


Kieran