Sean at Cosmic Variance wonders why we can’t visualize more than three dimensions. I find it both hard to imagine how you could visualize four dimensions and hard to imagine what biological feature of our brains prevents us from doing so.
Sean at Cosmic Variance wonders why we can’t visualize more than three dimensions. I find it both hard to imagine how you could visualize four dimensions and hard to imagine what biological feature of our brains prevents us from doing so.
My guess is that we can if we exercise enough.
It’s a poorly posed problem?
It’s a poorly posed problem?
It’s a pooly posed problem?
It seems to me that when we imagine things, we are artificially simulating our sense perceptions. I can’t think of any of our senses which are well-suited to model four dimensions (if anything most of our senses are less than 3d, e.g. hearing gives us something like 2 dimensional information). In fact, to see the 3 dimensional world we live in, we have to make plenty of assumptions already (which is why we are tricked by optical illusions). So in reality to imagine R^4, perhaps you don’t have to project merely down to R^3 but down to R^2.
Perhaps someone could come up with a strange game which decouples visual and auditory events to give an idea as to what R^4 “looks” like.
It seems to me that when we imagine things, we are artificially simulating our sense perceptions. I can’t think of any of our senses which are well-suited to model four dimensions (if anything most of our senses are less than 3d, e.g. hearing gives us something like 2 dimensional information). In fact, to see the 3 dimensional world we live in, we have to make plenty of assumptions already (which is why we are tricked by optical illusions). So in reality to imagine R^4, perhaps you don’t have to project merely down to R^3 but down to R^2.
Perhaps someone could come up with a strange game which decouples visual and auditory events to give an idea as to what R^4 “looks” like.
Note: I didn’t see the “reCaptcha” thing. So sorry for the re-post.
I thought seeing how everyday objects behave when projected into 4D and rotated would help build up an intuition of the 3-sphere and 4-space.
You can see some of the animations I made here:
http://spacesymmetrystructure.wordpress.com/2008/12/11/4-dimensional-rotations/
There are also some great youtube videos of 4-dimensional geometry.
But meanwhile..
– Jonathan Vos Post
=============
Polytope
The word polytope is used to mean a number of related, but slightly
different mathematical objects. A convex polytope may be defined as
the convex hull of a finite set of points (which are always bounded),
or as a bounded intersection of a finite set of half-spaces. Coxeter
(1973, p. 118) defines polytope as the general term of the sequence
“point, line segment, polygon, polyhedron, …,” or more specifically
as a finite region of -dimensional space enclosed by a finite number
of hyperplanes. The special name polychoron is sometimes given to a
four-dimensional polytope. However, in algebraic topology, the
underlying space of a simplicial complex is sometimes called a
polytope (Munkres 1993, p. 8). The word “polytope” was introduced by
Alicia Boole Stott, the somewhat colorful daughter of logician George
Boole (MacHale 1985).
The part of the polytope that lies in one of the bounding hyperplanes
is called a cell. A four-dimensional polytope is sometimes called a
polychoron. Explicitly, a d-dimensional polytope may be specified as
the set of solutions to a system of linear inequalities
mx =
5, there are only three regular convex polytopes: the hypercube, cross
polytope, and regular simplex, which are analogs of the cube,
octahedron, and tetrahedron (Coxeter 1969; Wells 1991, p. 210).
SEE ALSO:
Hypercube {below}
16-Cell
http://mathworld.wolfram.com/16-Cell.html
24-Cell
http://mathworld.wolfram.com/24-Cell.html
120-Cell
http://mathworld.wolfram.com/120-Cell.html
600-Cell
http://mathworld.wolfram.com/600-Cell.html
Cross Polytope
http://mathworld.wolfram.com/CrossPolytope.html
Face, Facet, Incidence Matrix, Line Segment,
Pentatope
http://mathworld.wolfram.com/Pentatope.html
Point, Polychoron, Polygon, Polyhedron, Polyhedron Vertex, Polytope Edge,
Polytope Stellations, Primitive Polytope, Ridge, Simplex, Tesseract,
Uniform Polychoron
REFERENCES:
Bisztriczky, T.; McMullen, P., Schneider, R.; and Weiss, A. W. (Eds.).
Polytopes: Abstract, Convex, and Computational. Dordrecht,
Netherlands: Kluwer, 1994.
Coxeter, H. S. M. “Regular and Semi-Regular Polytopes I.” Math. Z. 46,
380-407, 1940.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.
Emmer, M. (Ed.). The Visual Mind: Art and Mathematics. Cambridge, MA:
MIT Press, 1993.
Eppstein, D. “Polyhedra and Polytopes.”
http://www.ics.uci.edu/~eppstein/junkyard/polytope.html.
Fukuda, K. “Polytope Movie Page.”
http://www.ifor.math.ethz.ch/~fukuda/polymovie/polymovie.html.
MacHale, D. George Boole: His Life and Work. Dublin, Ireland: Boole, 1985.
Munkres, J. R. Analysis on Manifolds. Reading, MA: Addison-Wesley, 1991.
Sullivan, J. “Generating and Rendering Four-Dimensional Polytopes.”
Mathematica J. 1, 76-85, 1991.
Weisstein, E. W. “Books about Polyhedra.”
http://www.ericweisstein.com/encyclopedias/books/Polyhedra.html.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, 1991.
CITE THIS AS:
Weisstein, Eric W. “Polytope.” From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/Polytope.html
============================
Not Just Outside the Box, but Orthogonal to It
http://scienceblogs.com/principles/2007/05/not_just_outside_the_box_but_o.php
Category: Science
Posted on: May 25, 2007 12:28 PM, by Chad Orzel
One of the many after-hours events contributing to my
exhaustion this week was the annual Sigma Xi award and
initiation banquet, at which some fifty students were
recognized for their undergraduate research
accomplishments.
The banquet also featured a very nice presentation on
visualizing a four-dimensional cube by Prof. Davide
Cervone of the Math department here. He went through a
bunch of different ways to picture a four-dimensional
object through analogies to lower-dimensional objects.
It was as close as I’ve ever come to feeling like I
understood how to think about higher dimensional
objects.
In addition to that talk, he has a bunch of other
presentations online, as well as some nifty animations
of four-dimensional objects. It’s great, geeky fun.
Comments
# 1 | Jonathan Vos Post | May 25, 2007 07:25 PM
Tesseract: synonym for Hypercube. See:
http://mathworld.wolfram.com/Hypercube.html
Eric W. Weisstein. “Hypercube.” From MathWorld–A
Wolfram Web Resource.
Read, see pretty pictures, AND maneuver and rotate a
simulated tesseract with the mouse. Watch the
perspective shange it in fascinating way. Might give
you an aesthetic/kinesthetic appreciation of
hypercube/tesseract geometry!
Then click from there to other pages at Eric W.
Weisstein’s MathWorld… IMHO the best Math Pages on
the web.
“Polytope” is the multidimensional generalization of
polygon and polyhedron. It includes Tesseracts and
other things.
If you google “Polytope Number” you’ll find a page of
mine “Table of Polytope Numbers, Sorted, Through
1,000,000.”
Visualization of 4-dimensional objects is possible for
some people. It is extraordinarily rare to be “born
with it,” if possible at all. It can be learned with
varying degrees of difficulty depending on the age
that you start, how good you are at 3-D to begin with,
and how good you are at geometry and math in general.
One well-documented case is Alicia Boole Stott, niece
of THAT Boole, who invented Boolean Logic. He gave her
a set of colored blocks with instructions on what
colors could go next to what others. It was a toy
designed to get her visualizing 4-dimensional shapes.
She proved to be VERY good at doing so, into
adulthood. She could also go 5-D and 6-D to some
extent.
There was a burst of approximately 1,000 publications
about the 4th dimension in the late 1800s. This
forever influenced all nonfiction AND science fiction.
One Swiss gentleman who could visualize 4-D was Ludwig
Schlafli [umlaut over the a]
Born: 15 Jan 1814 in Grasswil, Bern, Switzerland
Died: 20 March 1895 in Bern, Switzerland
See:
Schlaflipage by St.Andrew’s University Math History
“Ludwig Schlafli first studied theology, then turned
to science. He worked for ten years as a school
teacher in Thun. During this period he studied
advanced mathematics in his spare time…”
He discovered something profoundly important. Let me
summarize:
* there are an infinite number of regular polygons,
like an equilateral triangle, square, pentagon,
hexagon… where all edges are the same length and all
angles identical.
* there are exactly 5 regular polyhedra, with all
faces the same and all angles the same: Tetrahedraon
(triangular pyramid), Cube, Octahedron, Dodecahedron,
Icosahedron. Everyone who plays Role Playing Games
knows these now as dice shapes. They are called
Platonic Solids.
* Schlaflii, home, alone, over a decade, discovered
that there are exactly SIX 4-D equivalents to Platonic
Solids, namely the Pentatope (4-D Simplex), TESSERACT
(a.k.a. hypercube, a.k.a. 4-D measure polytope),
hyperoctahedron, hyperdeodecahedron, hypericosahedron,
and one with no equivalent in any other dimension, the
24-cell.
* in all higher dimension, there are only 3: the
equivalent of terahedron, cube, and octahedron.
His work was published, but ignored, in part, because
so few could visualize. Alicia Boole Stott confirmed
his work: she could “see” it was true.
When I was a child, I learned to visualize 4-D
objects, in a hazy way. Later I became a
mathematician, and then a part-time professor of math.
Strangely, the visualization partially returned to me
at age 53. So I have written several math papers these
recent years, some devoted to 4-D and higher dimension
shapes. I’ve been emailing back and forth with real
experts, including Richard Feynman’s son Carl, about
the hypervolume of some of these shapes, and how that
is changed by truncateing them in various ways.
My son is mad at me for never being able to find out
the details of the colored blocks, and wishes he could
visualize 4-D. There’s a famous science fiction story
about a toy that teaches children to visualize 4-D,
recently made into such a bad movie that I won’t name
it, and they use to to sort of Tesser away.
I think a visualizer of 4-D is John Forbes Nash, Jr.,
the subject of “A Beautiful Mind.” His dissertation
was only 28 pages long. It was, formally, about
polytopes, but changed Economics forever.
“The King of Geometers” H.S.M. Coxeter died a couple
of years ago. His book “Regular Polytopes” [a Dover
paperback] helped me enormously as a child. The photos
and illustrations were beautiful.
Last month I was looking in a library at later
editions of that, and others of his books.
Now, much of those 2-D images of 3-D projections of
4-D objects are available on the web. I’d start at
mathworld.com, if I were you. In their search box,
type “pentatope” or “hypercube” and manipulate 4-D
objects with your mouse.
You’re not supposed to visualize it. It’s a mathematical model, not anything real. The closest you’ll come to “seeing” it is to become so familiar with its mathematical properties that you “get a feel” for a way to make “sense” of it.
ProfessorElvisZap on youtube draws the 1-skeleton of the hypercube on a blackboard. He uses a linear projection. The Kernel of this projection is 2-dimensional. So when we project from 4 to 2 we loose
2-dims worth of info. When we “see” only one dim is squished, and objects behind the visible objects are along rays. From 4-to-2 you loose planes.
Watching animations helps. Also, trying to interpret linear algebra geometrically helps. I won’t say I can SEE in 4-d, but by combining algebra, drawing, and analogy, I am pretty sure I have good pictures of the ordinary 4-d space. See also the drawings on the “math art” page above.
The following observation is made by Tim Gowers in ‘A Very Short Introduction to Mathematics’ I think:
If I ask: ‘How many edges does a square have?’, most people will either simply state the answer because it is learnt and known or they will picture a square in R^2 and count the number of edges. Most likely a lot of people will do both, just to be sure.
If I say ‘How many edges does a cube in 3 dimensions have?’, a much smaller number of people will have learnt the answer and most will picture a cube and count the edges. In our visualisations we sort of lay two squares on top of one another and then join the corners of the lower square to the corresponding corners of the upper square. Then we count how many edges in total.
Now, if I say ‘How many edges does a 4-D cube have?’, can we performe the same trick of visualising and counting? Yes we can! We put two 3-D cubes alongside one another and join all the corresponding corners. We can then just count the edges in our mind.
Perhaps it doesn’t feel like we are visualising 4-D space because we don’t know what 4-D space looks like…but you can’t deny that what we have is a visualisation that accurately tells us something about a 4-D object. If this isn’t visualising 4-D then what is?
Cube toys for that were invented by Hinton:
http://en.wikipedia.org/wiki/Charles_Howard_Hinton
Apparently that worked, e.g. Coxeter probably learned with them. A book on Howard Eves (”Eves circles”?) describes visualizations of 4- and higher dim. geometry by some famous mathematicians, if I remember corectly.
Did my submission have too many URLs to get through the filter? It’s a an area (hypervolume) in which I’ve been researching, publisging, teaching, and visualizing for decades. Please check the queue. The submission began:
====================
There are also some great youtube videos of 4-dimensional geometry.
But meanwhile..
– Jonathan Vos Post
=============
Polytope
The word polytope is used to mean a number of related, but slightly
different mathematical objects. A convex polytope may be defined as
the convex hull of a finite set of points (which are always bounded),
or as a bounded intersection of a finite set of half-spaces. Coxeter
(1973, p. 118) defines polytope as the general term of the sequence
“point, line segment, polygon, polyhedron, …,” or more specifically
as a finite region of -dimensional space enclosed by a finite number
of hyperplanes. The special name polychoron is sometimes given to a
four-dimensional polytope. However, in algebraic topology, the
underlying space of a simplicial complex is sometimes called a
polytope (Munkres 1993, p. 8). The word “polytope” was introduced by
Alicia Boole Stott, the somewhat colorful daughter of logician George
Boole (MacHale 1985).
Over evolutionary time, we’ve developed a visual system that is fine tuned for 2 and 3-dim objects. So, clearly these “physical” objects are the easiest to see.
However, we do “visualize” some pretty interesting abstract objects once we figure out how to express them in these terms, e.g., most people can see a hypergraph, using constructs that are easy to see in 2 or 3 dim. Incidentally, Jeff Weeks’ book called The Shape of Space is an amazing read on this topic, with very good aids to visualizing 4-dim space.
Hinton cubes, yes. I’ve conjectured for decades that a really good 4-D computer game would allow the children who play with it become effortlessly familiar with 4-D, and master Special and General Relativity easily.
I was quite influenced in this conjecture by Alicia Boole Stott,
about whom I’d read as a child. I worked hard at 4-D visualization
when my brain was still plastic. My son rebukes me for not giving him
the same advantage.
Alicia Boole Stott rediscovered the breakthroughs of the regular
polytopes that were discovered before 1852 by the Swiss mathematician
Ludwig Schläfli. He was not just a visionary Mathematician, but an
expert linguist speaking many languages including Sanskritt and
reading the Rigveda. His complete treatise – rejected by the Imperial Academy of Science – “is an attempt to found and to develop a new branch of analysis that would, as it were, be a geometry of n dimensions,
containing the geometry of the plane and space as special cases for n
= 2, 3. I call this the ‘theory of multiple continuity’ in the same
sense in which one can call the geometry of space that of three-fold
continuity,” was published posthumously in 1901, and only then did its
importance become fully appreciated.
Or, to give a short answer to the specific question: “I find it both hard to imagine how you could visualize four dimensions and hard to imagine what biological feature of our brains prevents us from doing so.”
A: None! Except that almost everyone starts much too late. You need to begin during youthful brain plasticity.
Nick says: “I can’t think of any of our senses which are well-suited to model four dimensions (if anything most of our senses are less than 3d, e.g. hearing gives us something like 2 dimensional information)”
On the contrary, most of us can spatially locate the source of a sound in 3-dimensional space quite accurately, and to do so in close to real time. (Having two ears helps with this.) In addition, those of us with musical training can ALSO, without any difficulty or even any apparent effort, determine further dimensions of sounds such as timing, pitch, rhythm, timbre, sound quality, type of attack, and expressiveness. As a violin student, I can usually tell whether two succesive string tones I hear are being played with slurred or separated bows, for instance. Similarly, most brass players can distinguish by ear music played on slide trombones from that played on valve trombones. MOREOVER, we can also relate these near-real-time determinations to past sounds we have heard (both immediately before and long ago) and/or to imagined future sounds. CAT scans of human brains have demonstrated that one pleasure we gain from music is a predictive one — upon hearing a sound, our brains anticipate the sounds which we expect to follow it, and are pleased when these mental predictions are borne out.
All this says to me that humans are able to model sounds in as many as a ten or more dimensions, to do so in near-real-time, and to do so simultaneously.
Jonathan Vos Post: Regarding the enormous attention in the late 19th century on understanding the fourth dimension — this focus bore fruit in early 20th century art, when artists such as Malevich and Mondrian explicitly sought to represent the (a?) 4th dimension in their art. See the following book for details of this:
@BOOK{henderson:book83,
author = “L. D. Henderson”,
title = “The Fourth Dimension and Non-Euclidean Geometry in Modern Art”,
publisher = “Princeton University Press”,
year = “1983″,
address = “Princeton, NJ, USA”}
I agree, Peter. That book makes a strong argument. And I agree with the multidimensionality of music, albeit I was a poor Classical Guitar performer, and merely profitable Opera producer. So “visualize” is biased towards one sense. In my teaching experience since 1973, roughly 40% of my students were Visual, 30% were auditory, and 10% were tactile/kinaesthetic. So musical multidimensionality can reach roughly 30%, and I contend that videogames can also reach the tactile/kinaesthetic 10% (aided by building and handling 3-D projectsions of 4-D shapes).
I like it that both Cambridge Philosophy Don/Science Fiction pioneer Olaf Stapledon and J. R. R. Tolkein described cosmoses where Music constitutes the dominant dimensions (Star Maker; and Silmarillion, where the war between angels and God was over harmony versus counterpoint).
Sure, once we all get Baby’s First Algebraic Transformation mobiles and Russell Blocks printed on the sides of polytopes. You know, easy dot algebras….
The examples; they just look like local group projections; how am I going to formally invert a horse through itself using negative lens distance in blender (or some other renderer)? (Tens of aliens tried it the tactile way; they can’t get it either. (cough)) It is too easy to confuse dimensions (which it would be our senses’ job to reduce to a salient few aspects), and plot coupons.
We barely have 3D animations of common particles; wasn’t it only recently that we got explanations of friction that weren’t cartoons of classical mechanics? So…how about just complex/real 3D visualization, now that we have (well…for $4k, wall space, and enough computer, videocard & webcam around to track whoever leads the ‘viewpoint’ around…carbon footprint flaming…hideous distortion of MTV Sound Fields through 2D comb filters…) 3D displays.
Perhaps if the atomic structures of the material that forms some hypothetical newly evolved cortex of the brain - if this atomic material were to contain an axis lying in a fourth direction; let’s say a .0000006 seconds into the future and .0000005 seconds into the past, and with particles at these distances, then information of four space might be able to be processed into perceptual interpretation mediated through the 4-D neurons of this cortex. -scriAlphi
As a visual systems neuroscientist I don’t find it odd at all that we cannot easily visualize four dimensional space. After all, our nervous system evolved in a three dimensional world (one we already have to approximate with fundamentally two dimensional data and computational machinery). You should be more surprised that we can accurately visualize three space.
“After all, our nervous system evolved in a three dimensional world” begs the question (both as the the actual dimension of reality, and to the content of geometric imagination). Or relies on circular reasoning. And can people who DO visualize four dimensional space be said to have evolved differently? With all due respect, visual systems neuroscientist Vincent Toups, you miss the point of the real question. Not that your implicit question “[how] can [we] accurately visualize three space?” is adequately answered so far by our field.
My old roommate once told me that her favourite shape was a tesseract. I thought it was a clever thing to say although a physics major acquaintance was critical of her ability to have a shape that she wouldn’t be able to conceptualize as her favourite.
There is a graph type called slice can show the character of 4D volume scientific data. The coordinate x, y and z are used to show the position in the volume, and the color shows the feature of the fourth dimension.
You can see some images made by visualization software Visual Data:
http://www.graphnow.com/visual-data.html
In the context of four dimensions, the general topological considerations confuse the concept. Taken from a combinational generalization, it is second nature to consider upwards of four to seven variables in state space. To simplify matters by considering known boundaries of a given variable range, in a similar relation to confining a axis of rotation in topological space. In addition, in many linear relationships the combinational permutations or ranges of all known state configurations, can be reduced to rate functions independent of forth dimensional time, and considered directly in context to their relations and boundaries of configuration. Visualizing four dimensions is a matter of context, when the ranges and boundaries are known many systems of equations can be reduced to very few if no dimensions at all.
In physics, time is the 4th dimension, and we move along the time axis to see how a 3D objects evolves ( for lacks of better word) —- or changes shapes. So if we have another spatial dimension, I wonder if we can use the same method? Slide along that 4th axis, and observes what happens in the other spatial axes.
“In physics, time is the 4th dimension,…” is how H.G. Wells could summarize an idea already popularized for over a decade. But that was assuming Euclidean flat space. Minkowski and Einstein rather changed all that. “Space of itself, and time of itself, melted way, and what remained was a kind of marriage.”
http://math.ucr.edu/home/baez/gr/oz5.html
“Now, what does geodesic deviation have to do with curvature, exactly? Well, let’s look at a spacetime diagram of the situation. Hang on while I equip you with 4-dimensional vision.” He passes his hands over Oz’s head and mutters an almost inaudible syllable, and all of sudden Oz is shocked to find that he can see the clocks, not just in space, but in spacetime. They trace out curves in spacetime, and he sees these curves in their entirety as a static whole! Yet he is simultaneously able to see how at any given “moment” — i.e., any given slice of spacetime — the clocks occupy positions in “space” just as they did in the previous scene! Furthermore, as before, he can see, as in a movie, how as time passes the clocks begin to accelerate away from each other. Somehow he is seeing not just space but spacetime!
“Hey, how are you doing that?” he asks. “I bet if I could do this myself these problems would be a whole lot easier!”
“I’m just using my limited telepathic ability to show you how I think of these things,” the wizard says. “For you to do it too, all you need is practice.”
Oz frowns, unsure as to how he’d practice this. He decides to keep quiet and pay careful attention; maybe he’ll get the habit of 4d vision.
“Wise move,” says the wizard, smiling.