As a physicist I can't help but notice the similarity of A = I + B/2 - 1 (27:35) and a hypercharge formula Q = I + Y/2 (where I is isospin and Q is a charge). On the one hand this is not really a big coincidence but then there is this half integer thing (30:10) which must definitely have connection with spinors in physics because that's a cross product and in Clifford algebra language it's a rotation plane (in prof Wildberger's terms directed triangle), and Clifford algebra is a go to language for describing spinorial objects, Pauli matrices generate Cl_3 and Dirac gamma matrices generate Cl_{1,3}. So I really wonder about the (27:35) formula.
Larger or equal to! Each edge has two faces, and say each face has three triangles (let's say it's all triangles). then 2E will triple count all the faces, so 2E/3 is the number of faces and hence 2E=3F. Now say we unite two triangles into a quadrangle by removing an edge, you have one less edge and one less face so 3F - 3 but 2E - 2 so now 3F is one less than 2E. You can go on removing edges and 3F will be decreasing faster than 2E.
Dear Prog. Wildberger, I have been following your class on algebraic topology with much interest. I was wondering; could you tell us what textbook, or other materials you use for your class? Thank you for the great lectures.
@jnwildberger Correction on blackboard at 7.50: A graph X is planar iff it does NOT (missing, but was said correctly) have K5 or K3,3 as a subgraph.
As a physicist I can't help but notice the similarity of A = I + B/2 - 1 (27:35) and a hypercharge formula Q = I + Y/2 (where I is isospin and Q is a charge). On the one hand this is not really a big coincidence but then there is this half integer thing (30:10) which must definitely have connection with spinors in physics because that's a cross product and in Clifford algebra language it's a rotation plane (in prof Wildberger's terms directed triangle), and Clifford algebra is a go to language for describing spinorial objects, Pauli matrices generate Cl_3 and Dirac gamma matrices generate Cl_{1,3}. So I really wonder about the (27:35) formula.
Could anyone please tell me at 4:30, why 2E must be larger than 3F?
Larger or equal to!
Each edge has two faces, and say each face has three triangles (let's say it's all triangles). then 2E will triple count all the faces, so 2E/3 is the number of faces and hence 2E=3F. Now say we unite two triangles into a quadrangle by removing an edge, you have one less edge and one less face so 3F - 3 but 2E - 2 so now 3F is one less than 2E. You can go on removing edges and 3F will be decreasing faster than 2E.
Very interesting - stuff I did not know - and then the stuff I did know was better explained.
Around 39:15 you posit the Pick's Theorem is for convex polygons only. Not true. Any closed planar figure will do.
Dear Prog. Wildberger, I have been following your class on algebraic topology with much interest. I was wondering; could you tell us what textbook, or other materials you use for your class? Thank you for the great lectures.
You would be surprised that still some universities students do not know about it.
Oh, We learnt Pick's theorem at year 5 in primary school. My teacher explained it in an easier way.
A nice visual derivation of the formula for a parallelogram (determinant formula): ua-cam.com/video/MreKwSgxMC4/v-deo.html