Re: HyperCube (Diamond Explained)

Benjamin Francisco P. wrote:
> 
> > ._____________________.
> > |\\.                 /|\
> > | \ \.              / | \
> > |  \  \            /  |  \
> > |   \  \.         /   |   \
> > |    \.__\_______/____L____\
> > |     |\. \_____/    _|_--""|
> > |     |  \L\.___L\.-" |     |
> > |     |   | |   | |   |     |
> > |     |  _|_|___|_|   |     |
> > |__.==|="__\L____\|=._|     |
> >  \    |    /       \   \    |
> >   \   |   /         \.  \   |
> >    \  |  /            \. \  |
> >     \ | /               \.\ |
> >      \L/___________________\|
> 
> > I always thought the above was a 2D representation of the 3D
> > representation of a 4D cube (hypercube), assuming there was a fourth
> > directional dimension... :-)
> 
> Actually, the above is the best representation possible for our senses,
> of a hypercube. The first method described (Creating 2 n-1 dim hc's and
> joinig ends works), points (0dim) become lines (1dim) lines become
> squares (2dim), squares go to cubes (3dim) and so on. Thwe trouble with
> all representations, is that, just like the previous 3, the joining of
> the 2 (n-1)dimensional hypercubes MUST be done at right angles to ALL
> the other dimensions used. In the first 4 (point, line, square, cube)
> that is just fine, cuz we can visualize the succesive dimensions. But we
> just CAN'T represent correctly a hypercube, because we can't imagine the
> 4th dimension, to draw the lines along THAT dimension (The weird
> hexagons drawn earlier are only 3d figures, because the lines joining
> the cubes are along one of the first 3 dimensions, not the fourth)
> The most accurate way would be for you to look at the drawing above, and
> try to imagine the lines connecting the 2 cubes being at right angles to
> all the lines they join. Tough huh? Well, don't worry, you CAN'T because
> we see the world in a 3d way, so you can't imagine a 4th d. Like you
> can't invent new colors.

Sure you can invent new colors.  Add a mixture of red, green, and blue
that no one has done before.

> In any case, I think a cube in our world is still a hypercube. Ok, so
> time is not the official 4th dimension, but it is still a dimension at
> right angles to all the others (of course) and cubes in our world DO
> continue on time, they don't just vanish. So they have an extention into
> another dimension (They probably extend into all dimensions, along with
> everything in the world, we just aren't equipped to notice it). Now try
> again, to imagine 2 cubes connected by lines at right angles to all the
> others. Do you understand why the cube above is still not an exact
> representation of a hypercube? After all, a 3d cube drawn on paper
> hardly resembels the real thing huh?
> 
> --
> I know the truth, and the truth hurts! -Isamu Dyson
> The end comes before chaos - Kefka
> 
> Benjamin Francisco P
> Bfr...@puc.cl

    A cube in our world is a hypercube only if its length along the 4-D
is equal to the lengths along the other 3 dimensions.  Otherwise, its a
hyper-rectangle.  Time is the "official" fourth dimension, according to
Einstein and physicists following him.  Einstein saw the universe as a
complete 4-D continuum, where you can "turn" it, and the fourth
dimension will take the place of another one of the dimensions -- kinda
like if you turn a regular cube, the width becomes the length or the
height depending on which way you turn it.  We can then represent a
hypercube after we come up with a way to reconcile the time/length
difference -- come up with a way to convert between meters and minutes,
for instance.  
    The other problem with the time-hypercube is the concept of world
lines.  World lines are lines on a regular cartesian plane, but with the
y-axis being replaced by the t-axis (t for time), that show how a
certain thing or set of things travel with respect to time and some
other dimension, like length.  An example is in order.  If you go in a
straight line for 2 meters and it takes you 4 seconds to do it (and you
did it at a constant velocity), then the graph would be a slanted line :


(secs)^
    4 +       *            
      |     /               
    2 +   /                
      | /                
  x <-+---+---+-->
      v   1   2  (meters)    
      y

with the equation :

y = (1/2 sec/meter)x 

World lines can be plotted for a hypercube, for instance, or all atoms
in the universe (on a *huge* graph).  The interesting thing is that if
you were to plot all atoms, then you'd have countless lines curving this
way and that, some of them coming together to form something, say a
hypercube, for a while, and then diverging when the hypercube is
destroyed.  But the lines don't begin or end, they are infinite in both
directions -- the graphical illustration of the law of conservation of
matter.  The other cool thing is you can plot energy as well as matter,
even on the same graph, because E=MC^2.  The way this is a problem for
hypercubes, is that if you plot it, all of the atoms inside are plotted
as lines with no ends, therefore the hypercube is plotted as a line (or
collection of lines, depending on how you want to do it) with no end.
therefore, a real cube with side = 2m plotted becomes a hyper-rectangle
with sides 2m,2m,2m,infinite seconds.

Have fun in ascii.physics land!

--
.--.            
|~~| .-------.  Kevin Kreamer
|~~| |.-----.|  <kre...@centraltx.net>
|--| || KCK ||  "Everything that can be invented has been invented."
|  | |'-----'|   --Charles H. Duell,U.S. Office of Patents, 1899. 
|__|~'-------'

Related ASCII art in the Gallery

Explore these categories from our collection of 11,000+ artworks:

   ____
 .'    '.
/'-....-'\
|        |
\.-''''-./
 '.____.'
       jgs

 _________
/_|_____|_\
'. \   / .'
  '.\ /.'
    '.'

Make your own ASCII art

Turn images, text or 3D into ASCII, or draw your own – right in your browser.