ITEM 4269604 (History of Science). Circa 1820, Sengai Gibon drew a famous calligraphy often referred to as "Circle, Triangle, Square": https://terebess.hu/zen/sengai.html Compare with the following algebraic plane curves: 1 = X^2 + Y^2 1 = X^2 + Y^2 - sqrt(4/27)*X^3 1 = X^2 + Y^2 - (1/4)*X^4 1 = X^2 + Y^2 - sqrt(4/27)*(3*Y^2*X - X^3) 1 = X^2 + Y^2 - X^2*Y^2 Did Sengai forget to draw a monogon and a digon? ITEM 4269605 (a minor gripe). The title "Circle, Triangle, Square" is not canonical. The same drawing is sometimes referred to as "The Universe". But how could one conceive of a universe without "Hotaru"? Did poor Sengai forget the Kanji for "Hotaru" as well? Or should we interpret Sengai's minimalism as a prognostication of future bio-devastation ( i.e. of "the impermanence of all conditioned phenomena" ) ? ITEM 4269606 ( On Impatience ). First, learn how to expand the following nomes: https://oeis.org/draft/A308835 https://oeis.org/draft/A308836 https://oeis.org/draft/A308837 After: https://oeis.org/A005797 The same technique applies to either of the following differential equations: 0 = (x-1)*(x-2)*T + d/dx( (x-3)*(x-4)*(x-5)*(x-6)*dT/dx) 0 = 15*(84 - 197*x + 78*x^2)*T + d/dx(4*(x-1)*(2*x-1)*(9*x-29) (9*x-2)*dT/dx) One is poorly-motivated, might as well be random, thus analysis of the nomes at any of the four singular points does not lead to much enlightenment. In particular, it does not even seem possible to integerize the expansion coefficients. As opposed, the second differential equation comes from a geometric precursor. It appears that the nome expansions around each of four singular points are integral after a scale transformation of the x variable: 2/9: 0, 1, 11742, 156203784, 2284549324448 ... 1/2: 0, 1, 304, 618436, 584357184 ... 1: 0, 1, 2152, 6629044, 21558540128 ... 29/9: 0, 1, 622056, 365686615476, 208699680038836576 ... The first one hundred terms have been quickly checked. PROBLEM: How to prove integrality? (not obvious to me). PROBLEM: WTF (What's The Function)? (Hint: Try searching through x-variate sphere curve geometries with local dihedral symmetry) PROBLEM: Once you've succeeded in finding something: Is it possible to use the nome variables to define pseudo-elliptic time parameterizations of the disjoint submanifolds? ITEM 4269607 ( On patience ). After many seasons of study in very dim light, a hotaru lantern takes definite form. Even more work remains to be done. PROBLEM: How do we construct useful objects in the material world? If we need to make a new lantern, what type of paper should we use? Cheers, Brad