I'm an incorrigibly geeky transboy with a love for neuropsychopharmacology, cacti (or plants and botany in general), weird music, spiders (and other cute animals), experimental film, might and magic, daggerfall, pokemon, and math (especially when symmetries are involved).
Symmetries of the tetrahedron.
This is by far the most lucid depiction of A4 I’ve ever seen.
I used to be a math major. Flipping through my old notes and… - Imgur
I tried making a snowdecahedron once but it wouldn’t really work out because this was in Arizona in late April so there wasn’t really that much snow so I just drew the graph of the dodecahedron’s vertices in the snow.
(Source: tavortiz)
0 (Zero) is the additive identity.
1 (One) is the multiplicative identity.
2 (Two) is the only even prime.
3 (Three) is the number of spatial dimensions we live in.
4 (Four) is the smallest number of colors sufficient to color all planar maps.
5 (Five) is the number of Platonic solids.
6 (Six) is the smallest perfect number.
7 (Seven) is the smallest number of sides of a regular polygon that is not constructible by straightedge and compass.
8 (Eight) is the largest cube in the Fibonacci sequence.
9 (Nine) is the maximum number of cubes that are needed to sum to any positive integer.
10 (Ten) is the base of our number system.
Some less arbitrary (as in human-dependent) facts about three and ten:
Three is the only dimension in which knots can be made. It is also the dimension in which spatial rotation corresponds to quaternion multiplication (and the highest possible dimension in which spatial rotation can correspond to a ‘nice’ set of hypercomplex numbers, as all the higher sets are nonassociative!).
Ten is the sum of the square of the first two odd numbers. Ten is also the number of vertices of the Petersen graph, one of the most important counterexamples in graph theory.
And some more:
Eleven raised to the nth power gives the nth row of Pascal’s triangle.
Twelve is the order of the smallest nonabelian alternating group.
Thirteen is the number of Archimedean solids.
Fourteen is the only Catalan semiprime.
Fifteen is the number of edges of the Petersen graph. It is also the first semiprime which is not the order of any nonabelian group.
In the academic world Dover Publications is widely known for publishing standard texts in mathematics. To me they’re known for publishing books with the best cover designs around, which truly make them stand out among the boring rest.
from my experience with Dover books, they tend to be very cheap, very short, and almost completely incomprehensible. don’t get me wrong, they’re very useful. and definitely worth the price. they’re just… not really introductory, given how densely they’re written.
My namesake the famous mathematical bridge vandal!
One proton, the most stable positively charged baryon, and one electron, the only stable negatively charged lepton. And of course the little one’s going to orbit the big one.
All the other elements just seem like accidents. As in, I could imagine a world in which carbon, iron, chlorine, etc. had not formed, but I cannot imagine a non-empty world obeying our laws of physics in which hydrogen had not formed.
There’s a possible reply to this, if this is taken to be an argument. The idea is that you can make inferences about yourself from what you can and can not imagine, but not about other things. It can tell you what you can and can not imagine, but tells you nothing of the status of imagined subjects. ‘I can’t imagine x’ is a psychological remark as opposed to something that can properly help us reach a belief about the subject at hand.
Apparently this is quite well accepted. (It’s a noteable counter response to one of Descartes’ most famous arguments.) I’m unsure about this response though. I’d like to think that boundaries of thought may seriously be instrumental to good critical thinking somehow. I have yet to think of a good response to this. (Although granted I haven’t spent much time on it.)
(I guess that, if we accept that once pre-big bang conditions were actual, then statistical certitude would be the proper way to infer the claim that hydrogen is a physical inevitability.)
Replying in particular to this bit: “I’d like to think that boundaries of thought may seriously be instrumental to good critical thinking somehow. I have yet to think of a good response to this. (Although granted I haven’t spent much time on it.)”
First I’m going to take it as axiomatic that math has something to do with critical thinking. So, there is a position in the philosophy of mathematics that states that mathematics essentially is a study of human intuition about particular concepts. In meatspace (in particular during discussions with other mathematicians) I’ve always heard it called nominalism, although the internet seems to call it structuralism or psychologism instead. I am sympathetic to this viewpoint (although deep down in my heart of hearts I’m a mathematical Platonist, as are probably 99% of mathematicians), and if you believe it, then math itself is founded on the structure of human thought. (This, of course, has problems with being culture-bound, but a lot of mathematical concepts tend to pop up independently; c.f. African fractal architecture which existed long before Western mathematics even knew about fractals.)
I once had a conversation with a logician (I don’t remember exactly who, but I’d put my money on it being Itay Neeman) in which he said that set theory is essentially the study of human intuition about infinity, ZFC is the codification of these intuitions, and what the plethora of forcing-derived independence results mean is that our intuition on this subject is really, really useless. So the idea that math is a codification of human intuitions is held in some repute among mathematicians. Therefore, if math is instrumental to critical thinking, then the hypothesis that the boundaries of thought are instrumental to critical thinking is actually incredibly viable.
As for the statement I made itself, I guess the “imagine” part was really more a statement about simulations. That is, if you could simulate the creation and growth of a non-empty universe, hydrogen would be guaranteed to appear in the simulations but other elements would not. But I don’t know enough physics to talk intelligently about that.
If you don’t read Sketches of Low-Dimensional Topology already, your personal astrologer advises that you start doing so before the next ecliptic.
Sample pics and text:
ctctstr4, originally uploaded by epsilon_is_afraid_of_zeta.
View this 3-manifold as an interval of concentric spheres where you have to imagine gluing the inner sphere to the outer sphere.
Near each point on a singular fiber, a regular fiber passes by some fixed number of times, the order of the singular fiber. In the picture above this number is 5 for both singular fibers.
Here they have order 2.
Here they have order 3.
Here they have order 1 and so they aren’t that special. A homeomorphism would make all the fibers appear as radial arcs, the S^1‘s of the S^1 x S^2.
For the conference honoring the 60th birthday ofCaroline Series(only a German wiki?!?), I was one of a handful asked to contribute pictures inspired by her work. First up is my contribution followed by a description. After that are a few more.
I don’t think I will ever be this awesome. Not only are the pictures way easier to understand than some scary symbols, but the text explains their meaning really clearly and in not-too-many words.
Pass the acid—I mean, the advanced mathematics—please. People thought Grigory Perelman was crazy for turning down a million-dollar prize and living like an ascetic. “Why should I jump for a million dollars, when I can control the vacuum space in between the quarks of the universe?” is my paraphrase of his reply. I don’t have a million dollars, nor do I understand all of this ring fiber link knot book page contact braid surgery stuff. But right now I’m honestly not sure which I would prefer: the imagination, or the dinero.
Thanks to Maxime (@2_43112609_1 on twitter) for the pointer.
I wish I understood all these images…
Does it still make an interesting structure?
I mean I know octonions are tied up with Lie groups in some way, but what about sedenions? And is this connection the same connection that ties S^1 to the complex numbers and S^3 to the quaternions?
