Geometry and Minimal Surfaces. Dr Giuseppe Tinaglia
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1 Geometry and Minimal Surfaces Dr Giuseppe Tinaglia
2 What is Geometry? geo = earth, metria = measure It all begun with doing measurements. Collection of empirically discovered principles concerning lengths, angles, areas, and volumes.
3 Thales of Miletus, 624-c. 546 BC Replaced experience with deductive reasoning. First true geometer. Thales Theorem,
4 Pythagoras of Samos, c. 570-c. 495 BC Very famous Pythagorean school. Fear of 2 = Let s do geometry.
5 Euclid of Alexandria, fl. 300 BC Organized much of what was known in geometry at the time in 13 books, the Elements. The Elements Masterpiece of logic Axiomatic Method. Most durable and influential book in mathematics. Second only to the Bible in the number of editions published.
6 Euclidean Geometry Assuming a small set of intuitively appealing axioms (rules of the game), deduce many other propositions. It is possible to draw a straight line from any point to any other point. It is possible to extend a line segment continuously in a straight line. It is possible to describe a circle with any center and any radius. It is true that all right angles are equal to one another. The troublesome parallels axiom: Given a line l and a point P not on it, there is one and only one line m through P which does not meet l. (Not very intuitive!)
7 Questions Is the Parallel Postulate really a necessary rule of the game? Can we change the Parallel Postulate into: there exists no parallel line or there exist more than one parallel line and never, EVER, incur into a contradiction? If so, then we have shown the existence of other geometries!
8 Founding Fathers of Non-Euclidean Geometries Gauss Lobachevsky Bolyai No parallel line exists, elliptic geometry. More than one parallel line exist, hyperbolic geometry.
9 Why was the discovery of Non-Euclidean Geometries a big deal? Immanuel Kant ( ) He said that space is objective, a priori, and it is Euclidean. The existence of Non-Euclidean Geometry strikes the first blow to this philosophy. General Relativity puts the final nail in the coffin.
10 Felix Klein ( ) Geometry: the study of the properties of a space that are invariant under a given group of transformations.
11 Differential Geometry Surfaces are studied with analytical methods. Bernhard Riemann ( ) Riemannian Geometry: Branch of differential geometry that studies properties of surfaces related to the measuring of distances.
12 The Theory of Minimal Surfaces Definition A surface M is a minimal surface if: M has MEAN CURVATURE = 0. Small pieces have LEAST AREA.
13 The Theory of Minimal Surfaces Soap films are minimal surfaces Plateau proved that they minimise area among nearby surfaces. Joseph Plateau ( )
14 Catenoid Image by Matthias Weber Proved to be minimal by Euler in Together with the plane, the catenoid is the only minimal surface of revolution (proved by Euler in 1744).
15 Helicoid Images by Matthias Weber Shape used by Archimedes to pump water in 250 BC. Proved to be minimal by Meusnier in Together with the plane, the helicoid is the only ruled minimal surface (proved by Catalan in 1842). The shape of a string of DNA resembles that of a helicoid.
16 Riemann minimal examples. Image by Matthias Weber Discovered in 1860 by Riemann, form a family R t, t > 0. Horizontal planes intersect these surfaces in circles or lines.
17 Schwarz Primitive triply-periodic surface. Images by Weber Discovered by Schwarz in the 1880 s, it is also called the P-surface. Such a structure, common to any triply-periodic minimal surface (TPMS), is also known as a crystallographic cell or space tiling.
18 Costa torus. Image by Matthias Weber Discovered in 1982 by Costa. This is a thrice punctured torus with two catenoidal ends and one planar middle end.
19 Costa-Hoffman-Meeks surfaces. Image by M. Weber Discovered by Hoffman and Meeks in 1983.
20 Genus-one helicoid. Images by M. Schmies and M. Traizet Discovered in 1993 by Hoffman, Karcher and Wei. Helicoid with a handle.
21 Genus-two helicoid. Images by M. Schmies and M. Traizet Helicoid with two handles. Discovered?
22 The Theory of Minimal Surfaces Joseph Lagrange ( ) Given a boundary, does there exist a minimal surface spanning it? This became known as the Plateau Problem.
23 The Theory of Minimal Surfaces In 1930, it was solved independently by Jesse Douglas ( ) Fields Medal in 1936 Tibor Rado ( )
24 The Theory of Minimal Surfaces What is the shape of a minimal surface?
25 The Theory of Minimal Surfaces What is the shape of a minimal surface?
26 The Theory of Minimal Surfaces Open Problem: Consider two curves on parallel planes. How many minimal surfaces are there spanning such curves? What is the shape of such a minimal surface?
27 The Theory of Minimal Surfaces Open Problem: Consider two curves on parallel planes. How many minimal surfaces are there spanning such curves? What is the shape of such a minimal surface?
28 The Theory of CMC Surfaces Definition A surface M is a constant mean curvature (CMC) surface if: M has MEAN CURVATURE = constant. CMC surfaces minimise area under volume constraints.
29 CMC surfaces in nature Soap films are minimal surfaces Soap bubbles are nonzero CMC surfaces
30 Delaunay Surfaces In 1845, Delaunay discovered and classified the surfaces of revolution with constant mean curvature H = 1. He wrote down a 1-parameter family of surfaces now called Delaunay surfaces (unduloids and nodoids). Such family contains a chain of round spheres of radius 1 and the cylinder of radius 1 2. They have genus zero and two ends.
31 Trinoid A Trinoid has genus zero and three ends. Each end is asymptotic to some Delaunay surface.
32 Examples of finite topology nonzero CMC surfaces Genus zero and 3 ends. Genus zero and 4, 6, 7 ends.
33 Smyth surfaces In 1993, Smyth classified all complete nonzero CMC surfaces admitting a one-parameter group of isometries. They are Delaunay surfaces together with Smyth surfaces. Smyth surfaces have genus zero and one end.
34 Wente Torus In 1984, Wente constructed the first example of a closed (compact without boundary) CMC surface in R 3 different from the round sphere.
35 The Theory of CMC Surfaces Given a boundary, does there exist a CMC surface spanning it? What is the shape of a CMC surface?
36 The Theory of CMC Surfaces Given a boundary, does there exist a CMC surface spanning it? What is the shape of a CMC surface?
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