## Download Algebraic Geometry. Proc. conf. Chicago, 1980 by A. Libgober, P. Wagreich PDF

By A. Libgober, P. Wagreich

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Complex Numbers and Geometry (MAA Spectrum Series)

The aim of this booklet is to illustrate that advanced numbers and geometry should be mixed jointly fantastically. This ends up in effortless proofs and usual generalizations of many theorems in airplane geometry, resembling the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem.

Additional info for Algebraic Geometry. Proc. conf. Chicago, 1980

Example text

Second proof. Let H and S be as in the first proof. Observe that H is homeomorphic to the space of unordered pairs of points u, v ∈ X. 6). Therefore, the surface S is homeomorphic to RP2 and cannot be embedded into R3 . This shows that H is self-intersecting, which implies the result. 4. Climbing mountains together. Suppose two climbers stand on different sides at the foot of a two-dimensional (piecewise linear) mountain. As they move toward the top of the mountain, they can move up and down; they are also allowed to move forward or backtrack.

Two mountain climbers. 5 (Mountain climbing lemma). Let f1 , f2 : [0, 1] → [0, 1] be two continuous piecewise linear functions with f1 (0) = f2 (0) = 0 and f1 (1) = f2 (1) = 1. Then there exist two continuous piecewise linear functions g1 , g2 : [0, 1] → [0, 1], such that g1 (0) = g2 (0) = 0, g1 (1) = g2 (1) = 1, and f1 (g1 (t)) = f2 (g2 (t)) for every t ∈ [0, 1]. The mountain climbing lemma is a simple and at the same time a powerful tool. We will use it repeatedly in the next subsection to obtain various results on inscribed polygons.

Use the mountain climbing lemma to parameterize the curves C ′ = { h1 (τ ), t1 (τ ) , τ ∈ [0, 1]} and C ′′ = {(h2 (τ ), t2 (τ ) , τ ∈ [0, 1]}, so that t1 (τ ) = t2 (τ ). Now define the average curve C ∗ of C ′ and C ′′ as C∗ = h1 (τ )/2 + h2 (τ )/2, t1 (τ ) , τ ∈ [0, 1] . By construction, curve C ∗ is continuous, starts at t = 0 and ends at t = 1. Consider a curve H given by the function h(t) = ϕ(u1 (t)). 13. Since h(t) is a continuous function, h(0) = 0 and h(1) = 1, we conclude that C ∗ intersects H at some t = T .