4D on Youtube
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kirby urner
2016年5月25日 下午1:13:352016/5/25
收件者：mathf...@googlegroups.com
(time more pronounced)
(many more on tap)
kirby urner
2016年5月26日 上午9:33:402016/5/26
收件者：mathf...@googlegroups.com
On Wed, May 25, 2016 at 10:13 AM, kirby urner <kirby...@gmail.com> wrote:
(Abbott's Flatland is critical to this subculture)
Abbott's Flatland was a work of political satire, not unlike
Gulliver's Travels (Swift). Intrinsic to the Coxeter.4D tribe
is a belief in "higher dimensional beings" that "really
exist", these beings being the Polytopes of Coxeter fame.
We're supposed to imagine ourselves as denizens of
Flatland and imagine how we might be confused by a
3D cube trying to talk to us. We wouldn't see it as it is.
By analogy, a 4D cube is outside our 3D experience.
How one imagines oneself as a being with no experience
of volume would seem impossible i.e. can we really
give up our sense of being inside a container with the
ability to contain containers? Does thinking of lines and
squares with talk balloons really get us out of that head
space? In science fiction, sure, it sounds believable.
In computer science we have multi-dimensional arrays
that provoke no superstitious beliefs in higher beings,
however once a metric (distance formula) is applied,
rendering the data makes it look coolly geometric
when "projected" i.e. there really is a way to slice through
a multi-dimensional array and get out polyhedra. The
data structures are coherent and model real phenomena.
The hypercube is indeed really a data structure. Belief
in "higher dimensional beings" in any other sense is entirely
optional but is pretty harmless, and helps add luster to
this corner of mathematics. "Higher dimensions" are
something to get religious about.
Remember 'Regular Polytopes' page 119? Coxeter
wisely wants nothing to do with time and Coxeter.4D is
completely time-free. The physics-minded tend to be
incapable of removing time from their thinking however
and may never grasp "multi-dimensional" in the way
Coxeter meant it. They should be forgiven as experience
minus time is about as anti-experiential as conscious
subjectivity minus either height, width or depth.
subjectivity minus either height, width or depth.
If you're into physics at all, you know about Special
and General Relativity, in which time is added as a
dimension, to so-called X, Y and Z. This then is the
"time is the fourth dimension" meme which is forever
confused with hypercube stuff. Science fiction encourages
this confusion, connecting "hypercube" to "hyperspace"
and thereby to "worm hole" and the ability to short circuit
the metric and pop out in a distant star system. This
is mostly just word salad, but it keeps readers amused.
Relativity is a confirmed phenomenon. Time dilation
does occur. A clock on a faster satellite orbiting the
Earth with tick more slowly relative to one that's at rest.
However the time dimension is not mathematically
treated as just one more spatial dimension as above.
Confusion is rampant, and I would say mostly deliberate.
Coxeter tried his best to keep these two language games
from colliding and becoming a train wreck, but he was
unsuccessful:
unsuccessful:
Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H.G. Wells in The Time Machine, has led such authors as J. W. Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.
H.S.M. Coxeter. Regular Polytopes. Dover Publications, 1973. pg. 119
see SynergeticsFuller.4D:(can't find any on Youtube yet -- haven't had time)
Only students of art history may be aware of Fuller's
use of 4D, which is neither of the above, though he had
great respect for both Coxeter and Einstein. Fuller was
interested in the primacy of the tetrahedron as a topological
volume, with four windows and four corners. He considered
it irreducible and did not encourage Abbott-like beliefs in
being with no depth, no height or no width (any of these
going missing is impossible in our experience).
Sure, there's restricting degrees of freedom, but any binding
of limbs by straitjackets or whatever are applied inside a
"room" or "cell" and the minimum room or cell has four walls
and four corners. He had a tetrahedral model of 3rd powering
Sure, there's restricting degrees of freedom, but any binding
of limbs by straitjackets or whatever are applied inside a
"room" or "cell" and the minimum room or cell has four walls
and four corners. He had a tetrahedral model of 3rd powering
to go with his tetrahedral (4D) beginning.
Although Fuller was an important architect and engineer,
his thinking was too philosophical for other architects or
engineers to wanna deal with it. Philosophers (in the sense
of employed academics with departmental responsibilities)
decided to leave it to art historians such as Linda Dalrymple
Henderson to remind us of this third meaning of 4D.[1]
Don't expect to find out much about it reading math books
as mathematicians have had little use for Synergetics directly,
though through osmosis much of Fuller's thinking has
percolated out into the culture. The octet truss and the
of employed academics with departmental responsibilities)
decided to leave it to art historians such as Linda Dalrymple
Henderson to remind us of this third meaning of 4D.[1]
Don't expect to find out much about it reading math books
as mathematicians have had little use for Synergetics directly,
though through osmosis much of Fuller's thinking has
percolated out into the culture. The octet truss and the
geodesic sphere are well known.
If you want more insights into Fuller.4D, I suggest reading
about Synergetics in Wikipedia.
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