tesseract

[papernuke]

To me a tesseract would look like a cube, but in the cube would be another cube (at the exact middle of the first bigger cube) then to the left, right, top, and bottom of the cube in the middle of the bigger cube which would be 8 cubes (that are all still in the bigger cube) like said in here http://tetraspace.alkaline.org/forum/vi ... =tesseract.

[moonlord]

That is incorrect. All the cubes in the tesseract are the same size, and all of them are sides of the tesseract. What you're describing is the projection.

[papernuke]

The projection? whats that?

[Keiji]

The projection is how an object looks in a lower number of dimensions.

[wendy]

Looking at a tesseract vertex first gives a rhombic dodecahedron. The eight faces appear separately as rhombohedra with the extra (pair of) vertices at the centre.

Still, what more can you say?

[danielmoore]

ive seen many different 2d projections of a tessaract but which one is most accurate. Are they all accurate, just different views? And what does a tessaract projection look like in 3 space?

[moonlord]

No projection is 100% accurate because it can't keep the angles. By definitions, a tessie has four angles of 90 degrees aroung a vertex, but you can only place three around a point in 3D space.

Every projection helps for visualising a certain aspect of the polychora (4D bodies). For example, a projection of the tessie shows two cubes of the same size, with their vertices connected respectively. This shows that you can create a tessie by expanding (extruding) a cube in a 4th dimension, perpendicular to all the previous three. The cube-in-the-cube projection shows how the eight cubic cells that are on the surcell of a tesseract are linked. And so on. Animated projections that involve the rotation in the XW, YW or the ZW planes are more difficult to interpret, but have patience and keep trying and eventually you'll succeed.

[wendy]

A tesseract can be projected to preserve angles, using the stereographic projection. You end up with a schlegel diagram, rather like soap-bubbles, and the most likely candidate of why the thing is called a polycell.

[moonlord]

I don't get it. I agree that stereographical projecting preserves the angles, but:

You can use stereographical projecting either on the 4D->3D projection or the 3D->2D projection, but if you apply it to both projections you'll get four images, and I guess most people don't have the required number of eyes...

[thigle]

wendy, do you mean by the soapbubble-like projection what Sullivan has on his page here ? :

http://torus.math.uiuc.edu/jms/java/stereop/

[bo198214]

And I wondered how he managed to get a smooth sphere in java2d!
But he is cheating! *ggg*
He made a picture (look here) of a smooth sphere and insert it into the interactive applet, because it anyway does not change under rotation.

[pat]

\You can use stereographical projecting either on the 4D->3D projection or the 3D->2D projection, but if you apply it to both projections you'll get four images, and I guess most people don't have the required number of eyes...

I don't understand this. If I use stereographic projections from 4D->3D, I start with a 4-D image and end with a 3-D image. If I use stereographic projections from 3D->2D, I start with a 3-D image and end with a 2-D image. If I did 4D->3D->2D, I'm not sure what step generated multiple images, let alone four of them.

[moonlord]

We might speak about different things. Aren't stereographical projections supposed to make two images, so they can be viewed by red-cyan glasses or by cross-eyeing or by parallel viewing? So every step duplicates the previous images. Isn't this the stereographical projection? :?

[pat]

Ah, no... those are stereopairs or stereograms. Stereographic projection is different.

[moonlord]

Lol.... Reading the thread again...

[wendy]

wendy, do you mean by the soapbubble-like projection what Sullivan has on his page here ? :

http://torus.math.uiuc.edu/jms/java/stereop/

That's the one. They tend to look like soap bubbles, because both preserve equal angles.

W

[batmanmg]

i think that all 2d (even if they look 3d) projections are almost completely incapable of portraying to the 3d mind what a 4d object looks like...

your simplifying the object too much... one must have a truely 3d rendering of the 4d objects in order to truely comprehend them and what they look like...

i can already hear you saying, the 2d image is preceived as 3d by our minds... thats true... but thats already a dimensional leep... we by nature have 2d minds becuase any picture we imagine is mimicing our 2d vision

i think the best way to beggin looking at a 4d object is to first understand how 3d vision works. (we have 2d vision so the next step up is ...) then you are really able to beggin imagining the 4th dimension.

[papernuke]

But then we can't see what it looks like, we could only imagine.

[batmanmg]

you don't see a 3d object when its drawn on a page either... you see a 2d image... and you imagine it as 3d... but you know how 2d vision works so this imagining comes as second nature...

you also can't see 4d... becuase you don't have 3d eyes... physicaly impossible for you... sry... you have to imagine it...

but imagining this is fairly difficult... i still barely can... but i've been getting closer after working wth how 3d vision works... its like a sphere expanding as it travels through 4d space... kinduf like how 2d vision is like a circle expanding as it travels through 3d space... making a cone shape. im not sure what to call the 4d cone is called.

im still working out perspective though... and how the vision interacts with other objects...

and yes i know that vision is light comming into the retna, and not something comming out of it... its just a simplifying rationalization of a complexe concept...