A Conversation for The Euler Equation

The Bridges at Konigsberg

Post 1

Kes

Is this the same Euler who proposed Euler's theorem in topology (the proof that it was impossible to navigate the 7 bridges of Konigsberg without retracing one's steps)?

He must have been a bright lad!


The Bridges at Konigsberg

Post 2

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming)

Oh he was. To give you some clue (he is my favourite mathematician smiley - smiley ) his nickname was Analysis Incarnate and he was known to dash off Proofs (including the Network Theorum you mentioned) between first and second calls for dinner.

He was also responsible for Euler's Phi function, Proof of FLT for n=3, Euler's Criterion, a proof of Fermat's Little Theorum and much much more.


The Bridges at Konigsberg

Post 3

Kes

This'll be old news to you - but it further confirms (for me) your opinion of Euler. I followed up your ref to "FLT for n=3" - took me a bit, being dim, but found a book titled "Fermat's Last Theorem" - a gripping read. It has a one-chapter bio. of Euler. Wow! He was indeed remarkable!


The Bridges at Konigsberg

Post 4

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming)

If that is the Simon Singh one then I agree its good smiley - smiley

I have it by me now smiley - smiley


The Bridges at Konigsberg

Post 5

Kes

Yes - that's the one. I've just reached the 20th century - ploughing on at a chapter a night.


The Bridges at Konigsberg

Post 6

Iacko

Kes: by the time i got round to replying to yr reply last week, you were offline and my computer crashed.

WARNING: I am not a trained mathematician, but tend to have some ideas that make some sense on some of the odder facets of math theory. I have a forum entry on my page about the star of david colour wheel and multi-D analogues which seem to fall into the Konigsberg bridge realm of maths (and oddly, it has piqued no interest.)


The Bridges at Konigsberg

Post 7

Kes

Thanks I'll walk across the bridges and take a look. BCNU


The Bridges at Konigsberg

Post 8

Iacko

here, or on the far side of the moibus?


The Bridges at Konigsberg

Post 9

Iacko

i'm not sure if you are thinking about this, or have wandered off somewhere else for the moment, but here is a bit more info:
Toss out Euclid's theorem about three point enclosing a plane. (If you wish, three points may DEFINE a plane; however, in order to solve this, planes must not be seen as obstacles or intersections.
If you'd like, it would only take me about three days to turn the raw film of these infinite construx into colour-coded photoshop/illustrator graphics.


The Bridges at Konigsberg

Post 10

Kes

Back again - just took a turn around the strip! I tried to visualise what you are talking about, and my poor old brain has problems - It's the sort of problem where an illustration helps. (Mind you, I haven't figured out graphics within H2G2 myself yet!


The Bridges at Konigsberg

Post 11

Iacko

Kes:
A serious inquiry:
As i have stated, I am NOT a mathematician by nature (*Paraphrases Shakespeare* geometry was thrust upon me.) Whilst in my final year at university, random ideas and arguments about minorly flawed points about accepted math and space theory all came together on a drive one afternoon. What I am talking about ties into Euler and Hamilton Circuits, Koch Curves, Sierpinksi Construx, Generalized recursive and symmetry theories, and owes a lot to the pioneering work of R. Buckminser Fuller. based upon my visitations to some of your fora and postings, i believe that you may be able to check some of my math and offer some input on whether this odd stuff that's been stuck in my head is merely a Borgean "Zahir," a mathematical oddity, or presents something of value to the field of advanced spatial theory.
Sorry to be so guarded, but, enough people think it strange that i am fascinated by an infintie geometric construx which i cannot explain to someone with less maths training than myself, and cannot describe to advanced mathematicians without getting lost; hence the reason i am multi-tasking and batch-scanning my negatives of this tiny little enormous thing as i type.


The Bridges at Konigsberg

Post 12

Iacko

Shall I surmise that you have lost interest in this geometry thingy, don't feel bad, everyone does. FYI: There is a group based in California of all places that calls itself the Fibonacci Society. The stated purpose is to investigate and discover natural and human functions of (Phi.) As i am certain you know, the Ancient greeks were way into this and the major Fifth is a major Phi function (cf. The Golden Section, published circa 1973 for both of the above statements.)


The Bridges at Konigsberg

Post 13

Kes

The geometry is a bit heavy-duty for my brain. I've read about the golden mean, and how it relates to music and heaps of other things - that I can grasp.
I think that these irrational numbers that keep coming up in all sorts of different areas are evidence that there must be some unifying rules that we haven't fully got yet, but that's about where my brain stops.


The Bridges at Konigsberg

Post 14

Iacko

Yeah, imagine being a Senior at University studying Byzantine History, Literature, and Popular American Culture whilst enrolled in a course on 20th Century Jewish Philosophy (the term/concept 'Theology' is anathema to the Jewish concept of YHWH--'He that cannot be named,' in one translation) upon which my degree in History rested, and stumbling upon a Cantorian infinite progression which resides in more than three dimensions, of which an incomplete model ('cos its infinite) can be constructed in our 3+1 existence using only slight modifications to Euclid's basic axioms and Plato's perfect forms, and which deviates from the generic (Phi) value by .015101261ish in terms of radii and relation to sidelength value from a 'TOP' to 'BOTTOM' polarity and which can arguably be seen as as the hypothetical basis for a TRINARY logic system and which seems to tie into some of the odder ends of R. Buckminster Fuller's speculation on form and function and the underlying wisdom of the universe. . .More or less, it's all about creating a self-replicating recursive Euler curcuit with three sets of mutually exclusive partners which cannot meet, but seem to meet at a posited intersection of the classic XYZ axis; however, since RED cannot be GREEN; nor BLUE, ORANGE. . ., it leads my mind into the thesis that the XYZ three-D cube model is a subfunction of a more perfect Hypersymmetrical Four-D construction. But, hey, i'm just a Photographer and imaging professional with a degree in History and such. Please forward this message and my email to anyone who might be interested. And I am also working on the illustrations.
My cousin is currently in Sydney, so i know the time is quite early beertime or perfect for T, so here's a cuppa what's good for ya.


The Bridges at Konigsberg

Post 15

Kes

Cheers!
OK - I'm still hanging on.
I can see that a 3-D object can be represented in various ways in 2-D - either by a "shadow" or an isometric drawing, so I can cope with the idea of a 3-D shape which represents a subset of the planes, vertices and edges of something in 4-D. Is there an agreed protocol for such representations?
(Here's where I start to loose it. I can cope with 3 spatial dimensions plus a time axis, but not 4 spatial dimensions).


The Bridges at Konigsberg

Post 16

Iacko

No worries, I've had to make it up as i went along. basically, a really simple idea turned oout to have much greater implications than i could have imagined. For a crossref. you might consider the tessaract (the 3D rep of a hypercube, immortalized in Dali's Christus Tessaractus painting.)

But here is is the quirk that I refered to:
The thinkers of such things think in terms of squares and cubes. This is understandable, as it provides a neat and orderly system; But the square is an innately unnatural form. (Witness the difficulty in devising 90 degree angles with a soap bubble computer.)
I can throw a brick, but at the end of the day, it is still a brick (never mind Xeno's arrow-- the ancient hint for Heisenberg's uncertainty principle), it is just a semi-imperfect cube that has traversed a semi-parabolic trajectory along XYZ coordinates, while the earth spins with the sun and moon and the sun spins with the Milky Way, and redshifting occurs overall. . .
Toss out the concept that three points enclose a plane.
Get there by disregarding that Euclidian Axiom as stated, replacing 'enclose' with 'define'.
Basically, what i am explaining here depends upon this crucial distinction.
i can justify the above via the existence of neutrinos, which bend this rule more times than i can count while i type this, or via the Sierpinski construction (EQ triangle with Zero Area amd an infinite boundary.)
Basically, the rules of simple geometry remain static, except that area and volume are disregarded via any means of logic deemed necessary by the thinker bound by thoughts along those lines.
Those lines and the joins of such are the important things.
Now we are getting into points in time that are in two places at once, depending upon your point of reference (The participatory Outside Observer conundrum.)


The Bridges at Konigsberg

Post 17

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming)

The Euler Phi function is a generalisation of Fermat's Little Theorum rather than being at all to do with the Golden Ratio. Just be pleased that Maths hasn't borrowed the use of norse alphabets yet smiley - winkeye


The Bridges at Konigsberg

Post 18

Iacko

The (Phi) that I ref. to is the 1.618. . . .. value ascirbed in American Texts. . .I understand that this is topically drifting; but this string on Hypersymmetry semed to find its home here, rather than other places.; however, since the formula in question does satisfy the equation: n(squared) - 1 = (n - 1)(n + 1), there might be some logic in its placement. Anyway, have you read the past postings from me (Iacko et. al)? Perhaps you can help me as well. A strange little huge infinite progression popped into my head on afternoon while careering down the motorway. I'm not a mathematician myself, but I've uncovered this one quite quirky recursive contruction which i am almost ready to nominate as my vote for the most simplistic and clear-cut model for a "hands-on" model of what String and (Cat)GU (T)heories might look like. Mind the difficulty in dealing with noncubic forms and functions.


The Bridges at Konigsberg

Post 19

Joe aka Arnia, Muse, Keeper, MathEd, Guru and Zen Cook (business is booming)

Hmm... get into contact with Joanna. She is the physicist. I know bits of number theory (hence the Euler phi function we studied for coursework at GCSE). One thing that annoys me is how different American texts are compared to International ones.

Phi is different to the phi function though... smiley - smiley

Could you describe the model in an article?


The Bridges at Konigsberg

Post 20

Iacko

I'm in the process of fleshing it out now. . .I would most likely need a bit of assistance in the number crunching, because i am of a more visual and spatially oriented mind, you know, little thingslike did I use the proper Tangent or cosine. hopefully, i should be able to produce a rough draft in the next 10 days or so, but the minutae of keeping my vehicle running, laundry, a few other freelance photo/graphic things that are postdue, pending or in the works, as well as a full work week are postponing this. It is really quite interesting, nonetheless, and i shall endeavour to work on this soon.


Key: Complain about this post

Write an Entry

"The Hitchhiker's Guide to the Galaxy is a wholly remarkable book. It has been compiled and recompiled many times and under many different editorships. It contains contributions from countless numbers of travellers and researchers."

Write an entry
Read more