Does Hyper-cube exist?

Jet2 · 05-25-2008

Or is it still just a theory?

Hypercube

Even after reading this simple explantion, I till don't quite grasp the hyper-cube concept. Appreciate if anyone can provide more clues to help me understand how it work and function?

Nootropic · 05-25-2008

Not in our three-dimensional world. Actually, any n-dimensional cube exists--in n-dimensional euclidean space. What that page is really showing is a projection of a hypersphere, seeing as how there's really no way to visualize a four-dimensional object with our three-dimensional brains. Dimensions greater than three are used all the time in mathematics and science, most notably in string theory.

Check out this article

C1ay · 05-25-2008

There's a 4D Rubik's Cube here you might enjoy playing with

There's also a 5D model here if you get done with the 4D model to quick

CraigD · 05-25-2008

Like it’s lower-dimensional versions,

the 1-dimensional line segment,
the 2-D square,
and the 3-D cube,

4-or-more-dimensional hypercubes (I always liked the catchier word for a 4-dimensional one, “tesseract”) exists as idealized mathematical concepts. Like other mathematical objects in more than 3 dimensions, you can’t easily make a physical object as an aid in visualizing them. Nootropic explained this nicely in his post, i think.

Fortunately, due to our experience with sketching 3-D objects as 2-D “shadows” on paper, it’s not to hard to extend this experience to get an intuitive grasp of objects in more than 3 dimensions.

To start, let’s consider what a line/square/cube/hypercube is, in the simplest language we can manage. All can be thought of as a collection of “corners” (vertexes) we can describe as coordinates using just the numbers 0 and 1.

A line segment is (0), (1)
a square (0,0), (0,1), (1,1), (1,0)
a cube (0,0,0), (0,0,1), (0,1,1), (0,1,0), (1,0,0), (1,0,1), (1,1,1), (1,1,0)
a tessaract (0,0,0,0), (0,0,0,1), (0,0,1,1), (0,0,1,0), (0,1,0,0), (0,1,0,1), (0,1,1,1), (0,1,1,0), (1,0,0,0), (1,0,0,1), (1,0,1,1), (1,0,1,0), (1,1,0,0), (1,1,0,1), (1,1,1,1), (1,1,1,0).

Corners that differ from one another by only one number are considered connected by an edge.

Next, consider how we sketch a cube on a sheet of paper. We could just sketch a square and explain that half of its corners are hidden behind the other 4

but this wouldn’t be very satisfying, so we rotate our point of view, first by shrinking some lines we can see and stretching ones we can’t

This still leaves lines hidden by lines, so we repeat it with some line we didn’t shrink in our first step

At this point, we could say we’ve sketched a tessaract and explain that half of its corners are hidden, but just as this wasn’t very satisfying when we did it with a square for a cube, we’d better shrink some more visible and stretch some more hidden lines to get

and again, repeat to get

You can keep this up for a 5, 6, or any number of dimensional hypercube – though eventually your page will be barely readable and nearly black with all the edges and corners!

Although lines, square, cubes, and hypercubes are mathematical, not physical objects, one can actually make pretty good actual wire-frame physical models of a cube, and actually rotate it to cast a shadow on a screen. Some “theories of everything” suggest that there may be additional spacelike dimensions, but most have these additional dimensions “compactified” in such a way that the whole universe in their extent is sub-microscopic, so the prospect of making an actual, physical model of a hypercube seems unlikely.

Jet2 · 05-25-2008

Thank you all for the explantation and references.
This is a good brain exercise.

CHADS · 05-26-2008

This may seem like an obscure way to explain it to a laymen becuase I am also the layman but here goes ....

The sketch you seen with the hyper cube had the red cube in the centre ....this is the 3d cube you used as reference and understand it .The other part of the hyper cube thats outside the red cube also looks like a 3d cube but bigger.

Notice the corners of the bigger cube have lines conecting to the smaller cube

This is becuase the Hypercube seen here is one dimension extra from a normal 3d cube so it is basically all the dimensions .... 1d .. 2d and 3d ... mashed togeather into 4d ..

So 1d ... is a line with two ends ..... 2 verticies . 1 edge(line)
2d .. is a square with 4 corners 4 verticies 4 edges
3d .. is a cube with 8 corners 8 verticies 12 edges
4d .. is the hypercube with 16 vertices. 32 edges

At each edge(line) in the hyper cube 3 cubes and 3 squares meet each other

2 cubes meet at each square

In each corner ; 4 cubes , 6 squares and 4 edges all meet up so turning that one corner changes all of these .... What a mess!

Basically the hyper cube is a cube shape object of eight cubes sharing all eight corners ... That sounds like one big fat cube with eight cubes squished togeather to make a heavy cube ... But all the cubes are attached to each other as quoted above so if you turn the overall hyper cube you see the other cubes pop into existance (as all the corners interact with 4 cubes 6 squares and 4 edges) ... they were hiding behind each other in 4 d space.

If you turn the centre red cube and it just spins around normally ... then it is not conected to the outer bigger cube or the 8 lines coming into the centre from the corners.

So Turning the red cube has to turn with the bigger cubes lines attached and attached in a 4d way.

When you turn the centre red cube you are turning the 8 vertices of the bigger cube also that are attached to 3 other cube shapes and 3 square shapes ... (sick i know!)

Becuase you cant see the cubes in 3d form its difficult but as you turn the cube its like seeing the other cubes and squares hidden like *inside out *teleportation ... This is four dimensions .. like an illusion in 3d.

Rubicks clocks !!! .. you move one and many more change . If you had a Rubicks clock that was a cube shape then you woudnt see what you had changed untill you moved the clock around to the back and looked ....; Similairly the hyper cube seems to hide parts of its self round the back.
Becuase our eyes cant see around corners (4d) ... !?unless we had a mirror?!

Hope this wasnt too patronising or more I hope its correct becuase as I said I'm a layman.........

Jet2 · 05-27-2008

So does it have hyper-globe?

CHADS · 05-27-2008

No a cube is a cube ....

The Hyper sphere is a globe and its volume is found like a sphere ...

Volume of 3d sphere .... = 4/3 pi r~Cubed

Volume of a hyper Sphere = 2pi~squared r~Cubed (there are 2 pi's here becuase of the extra rotation in another dimension) ( MISTAKE THIS IS THE HYPER SURFACE AREA)

The hyper cube has no circles ... spheres ... so dosn't need pi.

Unless you try to square the circle .. but there is a proof written thats states this is impossible...!!! You can never turn a square into a circle of any radius and use whole numbers ... You always have to use decimal numbers.

So im led to believe anyway!

Hypercube the hypersphere!

CraigD · 05-28-2008

Quote:

Originally Posted by CHADS

Volume of 3d sphere .... = 4/3 pi r~Cubed

Correct

Quote:

Volume of a hyper Sphere = 2pi~squared r~Cubed (there are 2 pi's here becuase of the extra rotation in another dimension)

Incorrect.

Your first clue something is wrong should be the units the formula returns. The “contents” (a general term for length, area, volume, etc. in units of any number of dimensions) of a 1-dimenensional object has units of $\mbox{lenght}^1$ , 2-D $\mbox{lenght}^2$ , 3-D $\mbox{lenght}^3$ , so a 4-D object should have $\mbox{lenght}^4$ units, not $\mbox{lenght}^3$ .

The volume of a tesseract (a 4-dimensional hypersphere) is: $\frac{\pi^2}2 r^4$
For a 5-D one, it’s: $\frac{8\pi^2}{15} r^5$

Here’s a list of the contents of hyperspheres from 1 to 20 dimensions:
$V_1= 2 r$ (a line segment)

$V_2=\pi r^2 \dot= 3.1415926 r^2$ (a circle)

$V_3=\frac{4\pi^1}{3} r^3 \dot= 4.1887902 r^3$ (a sphere)

$V_4=\frac{\pi^2}{2} r^4 \dot= 4.9348022 r^4$ (a tesseract)

$V_5=\frac{8\pi^2}{15} r^5 \dot= 5.2637890 r^5$

$V_6=\frac{\pi^3}{6} r^6 \dot= 5.1677127 r^6$

$V_7=\frac{16\pi^3}{105} r^7 \dot= 4.7247659 r^7$

$V_8=\frac{\pi^4}{24} r^8 \dot= 4.0587121 r^8$

$V_9=\frac{32\pi^4}{945} r^9 \dot= 3.2985089 r^9$

$V_{10}=\frac{\pi^{5}}{120} r^{10} \dot= 2.5501640 r^{10}$

$V_{11}=\frac{64\pi^{5}}{10395} r^{11} \dot= 1.8841038 r^{11}$

$V_{12}=\frac{\pi^{6}}{720} r^{12} \dot= 1.3352627 r^{12}$

$V_{13}=\frac{128\pi^{6}}{135135} r^{13} \dot= .91062875 r^{13}$

$V_{14}=\frac{\pi^{7}}{5040} r^{14} \dot= .59926452 r^{14}$

$V_{15}=\frac{256\pi^{7}}{2027025} r^{15} \dot= .38144328 r^{15}$

$V_{16}=\frac{\pi^{8}}{40320} r^{16} \dot= .23533063 r^{16}$

$V_{17}=\frac{512\pi^{8}}{34459425} r^{17} \dot= .14098110 r^{17}$

$V_{18}=\frac{\pi^{9}}{362880} r^{18} \dot= .08214588 r^{18}$

$V_{19}=\frac{1024\pi^{9}}{654729075} r^{19} \dot= .04662160 r^{19}$

$V_{20}=\frac{\pi^{10}}{3628800} r^{20} \dot= .02580689 r^{20}$

Hypersphere’s have lots of interesting properties, perhaps the most interesting one being that, for a fixed radius of 1, the numeric value of the contents reaches a maximum at 5-D. The mathworld article “hypersphere” has more information.

Quote:

Originally Posted by CHADS

The hyper cube has no circles ... spheres ... so dosn't need pi.

The 2-dimensional shadow, or any cross section cut by a 2-D plane, of a hypersphere of any number of dimensions greater than 1 is a circle. I’d say they have a strong circle-like quality to them.

CHADS · 05-28-2008

Sorry you are Right this was the surface area of a hypersphere!

Does Hyper-cube exist?

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