Question

4. a) Explain the following sentence: Klein space K3 contains a copy of the 2-dimensional Klein bottle K2in fact, it contains many copies). b) Explain the following sentence: Klein space K3 contains a copy of the 2-dimensional torus T2 (in fact, it contains many copies).

Answer 1

The five pieces of the triangle and of the lambert quadrilateral are matching as follows (a) a = Bc (2) b = AC (3) c = AB (4) alpha = angle BAC = pi(l*) (5) beta = angle CBA = pi(m) has angle of parllesim pi(q) = angle QRS = Ps = QR with l = PQ with m = RS (a) Klein space k^3 contains a copy of the 2-dimentional klien bottle k^2 and (b) klein space k^3 contains a copy of the 2-deimentional tones T^2 we use kleins disk model with disk D and choose center 0 = A = P The bijective correspondence of delta ABC and square PQRS is given by requiring the (i) A = P and c = s (ii) vertex B lies on the ray SR^rightarrow (iii) Hypothesis AB and Quadrilateral side QR have a common ideal end point v