Then you have "time as the fourth dimension" ala Einstein and the physics world, going back to the idea of "world lines" (scenarios) and rolling forward.
These two kinds of 4D are best encapsulated by two icons: the tesseract and the time machine.
Kirby, I do not know what time is more than anyone else, but the physical environment up and down, front and back, and side to side is pretty basic to our general understanding of direction as upright bipeds. How does that account for the vast amount of space we carry with us when we move through space? Or for creatures that fly, walk on vertical and overhead surfaces? Which now include some of us.They must work from a different axial symmetry than how we interpret. Time is taught, used, as a measurement to regulate extension through space. Time does not “roll forward”, our perception of consciousness moves. Time appears to have no obligation to space being an eternal state of transit experience. Space has been given us to know time. You are right about words we use, they are multifunctional to context, as all mathematics must be. And the context is never large enough to get a hold on what we are talking about.
This idea of Quadpods is suggestive of a different symmetry system. The so-called quadray coordinate system is useful for contrasting with XYZ, but also for comparing, as we have the same idea of vector addition, scalar multiplication etc. It's all very familiar, yet enough different to spark insights, or so one might imagine. Given the four basis vectors carving space into four quadrants (vs. XYZ's eight octants) you also have another handle on this alternative meaning of 4D ("arrowhead geometry" one might call it -- links to the Lakota "four directions" for some readers, although that was planar).
http://controlroom.blogspot.com/2009/03/quadpod.html
http://en.wikipedia.org/wiki/Quadray_coordinates
I do not consider the tesseract/hypercube or time travel to have anything to do with the 4th dimension, unless we call the 4th dimension imagination, best considered as movement of relationship between known and unknown. That I suppose would be a better fit than time.
I was more just doing some intellectual history. This 4th dimension stuff was getting off the ground with a lot of people around the end of 1800s. No account is complete without some mention of Hinton, also P.D. Ouspensky. Linda Dalrymple Henderson does a good job with her art history.
http://www.finearts.utexas.edu/aah/art_history/faculty/henderson.cfm
http://www.worldsciencefestival.com/linda-dalrympe-henderson
http://en.wikipedia.org/wiki/Charles_Howard_Hinton
As for what the 4th dimension "really is", I wasn't doing much pioneering in my last post, just diagramming three "namespaces" as I call them. The usual two (that some might confuse for one) are among the ones that survived the shakeout, one might say, of the early 1900s.
These would be the tesseract and the time machine (good book title). I peg them to Donald "King of Infinite Space" Coxeter and Albert Einstein respectively, but they're really camps, with whole groups of curriculum writers in each.
Then this meaning of 4D relating to the tetrahedron, with Frequency an added dimension (relating to time) came out in the late 1970s in two volumes, with dust cover blurbs by Arthur C. Clarke (science fiction), U Thant (UN), and several others. You seem to be already familiar with this history. I was just finishing at Princeton at the time and starting to teach high school geometry, calculus and world history. Having studied Wittgenstein quite a bit, I was intrigued by all these different usage patterns, what I called "re-vectorings of key terms" (like charged particles in a field).
These tomes were in collaboration with E.J. Applewhite, an interesting character in his own right and a friend (I sent him lots of writings, including from Bhutan). He also collated a 4-volume dictionary out of 5 x 8 note cards, trying to capture the usage patterns associated with this namespace. This is now available on-line thanks to Dr. Bob Gray in New York.
In my Coffee Shops Network blog (where I plot ways to bring colorful geometry cartoons to studious thinkers in student unions, at places like MIT), I write about 4D vs 4D vs 4D, another way to underline that we're dealing with multiple trajectories, multiple lineages. Our group at the Linus Pauling House was fortunate to have Dr. Mario Livio as our guest, author of some interesting math-related books, such as a recent one on Phi.
He listened to me practice giving a talk on this subject, in preparation of a 3 hour workshop in Chicago, where I was co-presenting with Steve Holden, chairman of the Python Software Foundation (we've since collaborated on other projects).
Here's a blurb about Dr. Livio's visit, with pictures. Glenn Stockton also gave a presentation (I hope you'll forgive all this chronology here -- just wanting to give some background and texture to where I'm coming from with all this curriculum writing -- I'm into integrating time lines, as you know). I've been mentioning Glenn to Milo, as they both seem deep scholars in ancient mathematics, plus have training in languages and cryptography (Glenn was with NSA for many years, a code cracker).
http://coffeeshopsnet.blogspot.com/2009/02/glass-bead-game.html
They know what it means to use the same word to mean different things -- comes up in real life all the time.
“Real life” is a strange concept; it comes up frequently in math talk, and in other frames. That term confuses by suggesting there is life that is not real, unless you mean “virtual”, life which is used to escape having direct interaction with other forms of life. We have lost any meaning to life when we confuse it with machines, all the way down to the electrical chemical level. Our ego has gotten so much larger than our capacity because our technological advancements have become ill-proportioned and out of balance to cultural achievements, thereby diminishing our human potential. So much life and we are forever finding ways to limit our interaction to mind concepts thereby cutting ourselves short on physical experience and moral sensitivity.
I sense a hunger, including among the technologically sophisticated, to reconnect with the natural world, and to rescue said world. What I write about on the Math Forum all the time is a mathematics curriculum that takes you outdoors. In the city, it might help you learn the public transportation system (which might encourage said systems to improve). GPS is involve (geocaching is a sport).
I call it variously "off your duff math", "girl scout math", "army math" depending on my audience.
I also write about First Person Physics and have gotten some kind attention from one Dr. Bob Fuller on that one, an influential professor emeritus based at University of Nebraska at Lincoln.
This approach to physics is likewise about using your own body, learning about calories and joules, and the concept of work (energy), not just by reading about it or working with weights and pulleys, inclined planes, but by working your heart and lungs, riding bicycles, measuring intake and output, studying the biochemistry involved, studying medical devices and instruments and the generalized principles behind them (MRI, PET scan, etc.)
http://physics.unl.edu/~rpeg/First_Person_Physics/urneressay.html (old essay, 2003)
Physics is becoming more supportive of medicine, the health sciences, over the long haul.
I was confirmed in this prediction at the recent AAPT conference (Association of American Physics Teachers) here in Portland, where I shadowed Dr. Fuller for a couple days.
The days when physics was about endangering the planet with radio-toxins is likely about over. We're in "countdown to zero" mode now (i.e. eliminating nuke weapons, if not nuke energy).
http://controlroom.blogspot.com/2010/11/more-global-data.html
http://worldgame.blogspot.com/2010/07/countdown-to-zero-movie-review.html
Thanks, had not seen the wing flapping video before. No it is not like that analogy of a bird flapping and flying in circles at all. We can say folding circles is like origami because they both use paper and you fold it, but you will only know the difference by folding. The mind has nothing to gauge worth if there is no experience. If words and other symbols do not simulate action, then they have no value. That is the risk we take in writing anything; there is no risk in doing something.
I'm likely going to understand your circles stuff better when I get to see it in action, like on Youtube or live.
I've been excited to find some of your writings dovetailing with mine and I see we've both explored some of the same literature, so that stands to reason.
In philosophy, we're used to individuals taking several volumes, lots of writings, to evolve their respective namespaces. It sounds to the casual ear that we're all speaking the same language, but when you encounter deep thinkers, you realize they've usually gone off the deep end into some rather private / original way of wiring stuff up (a unique namespace).
Getting your mind wrapped around someone else's usage patterns, even if the words are familiar dictionary words, is no easy task. However, given sufficient resources and media, such a project is doable. This is what goes on in the humanities all the time of course. It's also what goes on in mathematics, in physics.
Having shared mathematical concepts and symbols is of course a big help (potentially).
I think that's a long way of saying I encourage you to continue being verbose about your circle geometry and its curriculum. Keep hammering away on your namespace, as I've seen you doing here on mathfuture, with others following along. I'm sorry if I seem dense and slow -- we might call it the Doppler Effect although for some that might just add to the noise.
Your meanings will gradually resolve for some readers, in proportion to your ability to spell it out and get it out there. As Maria was saying, lets not tell ourselves that what's difficult is actually easy, when it's actually difficult. It takes time to develop one's meaning of "time". No one learns to play dynamite chess in a day either, unless they're a freakish prodigy of some kind.
Kirby
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