The slide at 43:08 is wrong. When you start to work it out you immediately see that the denominator must be way larger, and in fact the continued fraction is equal to 103993/33102. The slide is correct if you erase the 1/292 term. In general, to get a good rational approximation from a continued fraction you want to stop right *before* hitting the big number, not after like prof Baez says.
Always joke around and say 5 is self centered lol 😂 it’s the only number that doesn’t have an opposite in multiplication and it’s the center number between “1-9”
Another place to find fives is Mayan math/arithmetic which uses a base 20 number system subdivided into horizontal bars of five and dots above the bars as ones. 5 fingers x 2 + 5 toes x 2 = 20 - hmmm?
No, in all dimensions above n=4 there are only 3 regular polytopes.. the simplex, and cross, both made of tetrahedra and ultimately triangles, and the hypercube family made of cubes / squares. The exceptions to this in lower dimensions are the pentagon based figures found only in dimensions 3 and 4, their duals based on triangles, and the truly unique 24-cell in 4-d made out of octahedra. marcelteun has some cool heptagon based polyhedra, though they are not regular.
Also note how pi, a transcendental number, has very good rational approximations from its continued fraction. The numbers in the continued fraction of e are 2,1,2,1,1,4,1,1,6,1,1,8,1,... so you'll find very large numbers and thus very good rational approximations. Square roots on the other hand have periodic continued fractions so the numbers are bounded. Rational numbers themselves are the hardest to approximate with (other) rational numbers, so I would say that phi is actually the most *rational* irrational number.
32:28 My take on this is that they were turning fractions into unit fractions (virtually the only fractions used by the Egyptians, see "the rhind mathematical papyrus, an ancient Egyptian text") by use of continuing fractions.
Mind blown. I get the 4th dimension now... sort of. The 120th dodecahedron is the space outside the drawing, essentially everything, including us. To travel into the next dodecahedron in 4d space would flip the edges around, making it look like we're outside the drawing again, and so on.
Very interesting and worthwhile lecture. Regarding the stone geometric objects from Scotland mentioned early in the lecture, Michael Atiyah described them in a short book on Physics and Geometry. Sir Michael could only speculate on their significance, but his book has nice photos of the objects. Regarding the Pythagoreans, although they professed vegetarianism, they were forbidden to eat beans. A rather strange rule, and nobody is sure why they had this taboo. Excellent lecture.
And it all comes back to Pi :) There is something deeply eerie about how this relates to Kepler's Mysterium Cosmographicum. It looks like he was on to something, but in a way he simple couldn't imagine at the time. Seeing the dodecahedra laid out in 4 dimensions is just beautiful, but it kind of makes sense that the 120 cell comes from taking the 60 unique representations in 3 dimensions, up to the 4th. Like the number 5 in 1 and 2 dimensions, the dodecahedron is out of place in 3 dimensions, but it seems at home in 4. It truly is a rebellious albeit awkward number. I didn't realise that the golden ratio emerges from 5, and the fact that replication of cellular life in 3 dimensions is fundamentally tied to this it really makes you wonder how many dimensions we truly live in :) Fantastic presentation!
It's humorous at 46:25 where he attempts to demonstrate what he calls a "5th" chord. Of course, a 5th is an interval not a chord - which requires a minimum of 3 notes. Anyway, he then goes on to arpeggiate a major scale and ends up at the octave. Which is technically an 8th interval, not a 5th (or a 6th).
I can't get over the non-standard pronouciation of the greek alphabet (pi, chi, psi, phi, xi should rhyme, beta, zeta, eta, theta rhyme). How can a mathematician not have learnt the greek alphabet phonetically? Strange.
That's how it's pronounced in Greek, so it's actually more correct than your version. Don't like it? Maybe you should worry about how to spell "pronunciation" first.
Let me see... "Even though he was an engineer, buckyballs were named after him." Imhotep was an engineer. Leonardo da Vinci was an engineer. Nicolai Tesla was an electrical engineer. I'm afraid this sounds a little patronizing to me, since one of my hats is the engineer hat. My other hats include being retired, a Vietnam Era Veteran, and the author of 29 first-authored or single-authored journal articles on the mathematics of tomography and image restoration. And you should have gotten a laugh after your dodecahedron remark, especially with the Scottish audience. I don't know how you'll take this but please try not to look so elitist. Nowadays everyone who belongs to an elite tries to look like a hippie. It's a vanity, really. Real hippies don't look like hippies anymore. They tend to resemble Walter White, if anyone. But thanks for the information about Pythagoreans, and the golden ratio. That I like.
*5 = Grace and 5 = Enmity that is in the middle BETWEEN Them and US*.. The number 5 in the Bible is significant because his creation, the ‘man’ has five fingers, five senses and five toes. Thus it is the number of God’s grace. There are five great mysteries: Father, Son, Spirit, Creation and Redemption. After the fall of man creation was cursed and it became subject to vanity. So man and creation needed to be redeemed therefore number 5 is the number of God’s grace. The number five may also speak of the inability of man and his weakness as only when ‘man’ is weak does he needs God’s Grace. As only if a man is incapable, he would require God’s Grace. Thank You for this teaching I enJOYed it.
Except that there are thought to be 7 senses, not 5. In addition to the standard 5 everyone knows, there are also vestibular and proprioception. And I wonder if there are others we don't understand yet (thinking about how there seems to be like a psychic tether between some people, where one can think about the other and then the other calls out of the blue, things like that). Maybe there are 8 senses haha, that'd be funny.
@@melanierobson3336 wow i commented that 3 yrs ago.. now I see... pull up that page Proverbs 3:5-6 ( thanks for reply ..that was a blast form the past!) on FB I my group/page is Fig Informer check it out .. God bless
