Acme

The Fractal Geometry of Nature by Benoit Mandelbrot @ Amazon The Fractal Geometry of Nature @ wiki

Noting again the OEIS error in listing {1,2,3,4,5} with the set of non-polygonal numbers, I want to point out that while 3 is technically the second triangular number (and 1 is the first), it is proper to exclude them and all n= 1 or 2 values for all polygonals of side s. Otherwise, all numbers would be trivially polygonal. OEIS does properly show this restriction but does not follow it with the listing. Just so, and following the restriction, it can be proved that all multiples of 3 are polygonal and so no non-polygonal numbers are multiples of 3. Here's my proof. [math]F=frac{1}{2}(n^2s-2n^2-ns+4n)[/MATH] let n = 3 let s = {2,3,4,5...} set of integers >=2 [math]F=frac{1}{2}(3^2s-2*3^2-3s+4*3)[/MATH] [math]F=frac{1}{2}(9s-18-3s+12)[/MATH] [math]F=frac{1}{2}(6s-6)[/MATH] [math]F=(3s-3)[/MATH] [math]F=3(s-1)[/MATH] Δ, all multiples of 3 are figurate numbers Δ, no non-figurate numbers are multiples of 3 (sorry for the latex errors; little help?)

Just dropping in to drop off some eye-candy. The design is mine, the execution is computer generated by fella_2 and recolored by hand by moi. Not sure if spiral arrays are original to me, but I came up with the idea independently. The array begins in the middle, spirals out clockwise, and sequential cells represent the natural numbers. In this particular view the black cells are polygonal numbers and white not. Thanks for having a taste & hoping it's as fun to eat as it was to cook.

Excellent!! Thank you!! This has been a long-standing question for me/us. Here is my friend's version of a solution and another fella's variation for comparison. I will pass this all on to them as they came about the solutions independently. Coinikydinkily, the variant example I'm giving uses the same number 325 as the example on page 259 in the track you gave. I will be looking at the geometric analysis a bit more as we came at things only algebraically from the generalized expression for polygonals. Fella 2 has also been kind enough to do some programming for me and set me up with portable Python so that I could generate sequential extensive lists of the polygonals which I have done on the interval 6 to 1000000000. In that interval I found the greatest "multiplicity" to be 17 at the number 879207616 with the following solutions [n,s]. [4, 146534604, 7, 41867031, 14, 9661624, 16, 7326732, 28, 2325948, 31, 1890771, 56, 570916, 58, 531888, 118, 127368, 236, 31708, 248, 28708, 406, 10696, 496, 7164, 518, 6568, 1711, 603, 2146, 384, 7192, 36] This is all rather incidental to my interest in the set of numbers which are not polygonal and perhaps I could impose on you further for any information/knowledge you might have of that set. ? The OEIS listing of the set gives credit for the set listing to a fella named Beiler but Beiler's cited work we have checked and found it does not give that set at all. (OEIS actually gives the set incorrectly in listing {1,2,3,4,5} as members in spite of the restrictions n & s >=3.) Anyway, this is a recreational pursuit of mine that keeps me off the streets and I sincerely appreciate the help and interest. If you do have an interest in the non-polygonal set, I have independent work o'plenty to share. Thanks again.

Thanks. I see the volume goes for $92.00 as an e-book so it could be some time before I could purchase it as I am retired & on a small fixed income. Google books gives only a few excerpts and they do not address my question. I doubt my local community library has it but I will check. My community has no universities either, though we do have a community college which I can check too. Do you know for a fact the solution(s) I seek are in the book? Do you have the book and if so are you willing to check it for me? Do you know of a solution or if one exists? So as not to appear too "dodgy" I should say that I do know a solution though it is not mine. It can be written in a short paragraph and does not involve any complex mathematics at all. What I want to get at is if there are any other extant solutions and if so who wrote them and when and how they compare to the solution I have. You [all] of course understand my reticence to believe Fermat's claim and I beg everyone's forebearance in not giving the solution I have for the time being. In part I want to avoid presenting a bias and while I do have permission to use it I would want to touch base with the author again before posting it here. (Assuming anyone is interested in such a thing.) Thanks again.

Dato numero, invenire quot modis multangulus esse possit, or Given a number, to find in how many ways it can be polygonal. I am looking for historical solutions to this problem of Diophantus, whether old or recent. I have found that Fermat implied he had a solution, but he gave no actual solution. (as if. lol) Any help is appreciated. Thanks.

How did I miss that!? I was taking the original question at face value. My mistake & thanks for the heads-up. Marking topic ultra-super-dooper crackpot nonsense. And I thought I had joined a science forum. D'oh.

In my opinion, the ice analogy is a poor model for plate tectonics. The OP is quite specific and i found all the replies lacking so I gave specific examples of subduction under oceans. Then too, not all spreading centers are oceanic, e.g. the East African Rift and the molten basalt is a result of the hot asthensophere rising and decompressing so the basalt is not the driver. The production of any igneous rock is more of a chemical matter than it is mechanical.

Again, not all subduction zones involve continental plates. For example, the Mariana islands are ocean volcanoes along a line where the Pacific Plate is subducting under the Mariana plate. Neither of these plates is continental. Mariana Plate

subduction does occur in the [middle of] oceans. Plate Tectonics: Island Arcs

You're welcome. I recommend buying & reading the book as snippets here & there do not convey the whole of the ideas and arguments concerning I. Be forwarned it is not the kind of reading for persons of small souls. Enjoy.

I don't know the answers to the snipped bit, but I do know something on the add on question in regard to bacteria in the human gut . I hope a wiki article will suffice; if not there are other links there to more in depth expositions. : Gut flora

In his book I Am A strange Loop, Douglas Hofstader draws a downward pointing cone as a graph for self-awareness. the lower on the cone the less "soul" [say self awareness] a creature has. he labels the unit value "hunekers" after James Huneker who wrote a line containing the phrase "small souled men". Hofstadter rates dogs as having fewer hunekers than people, but more than mosquitoes. one has to make their own assesment about where each creature lies. the graph reads the same in a mirror, which would definitely please Hofstader even if he did not intend it.

No, inbreeding does not necessarily lead to defects. Here is a chart from Wikipedia of some probabilities. Inbreeding

I-racked my brain and I thought of Conway's Life. Hope that's not off the topic but I hope it does stimulate questions. For me the garden in the Pentateuch is no more or less interesting than any other ancient myth, but I do find them mildly interesting. What will we think of next? http://en.wikipedia....lular_automaton)

Every dog is an island.

Greetings. I am Acme and I joined to answer a question on identifying a rock. I have a modicum of scientific knowledge and I am the primary supplier of industrial equipment for coyotes. Looking forward to satisfying my wants and meeting your needs.

These are not meteorites. The rocks are hematite, an oxide of iron. Hematite