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4/13/2013 8:14:10 PM

Isoperimetric Theorem and Inequality

Isoperimetric theorem has been known from the time of antiquity. Elegant all by itself, the theorem also has a fascinating history. The most important contribution towards its rigorous proof was given in 1841 and is due to Jacob Steiner (1796 - 1863). At that time, it stood as the center of controversy between adherents and analytic (i.e. using Calculus) and synthetic (pure Geometric) methods. Although accepting the validity of the analytic methods, Steiner only used the synthetic approach. His proof contained a flaw that was later fixed by using the analytic apporoach. To better understand the nature of this discussion, let me first formulate 2 equivalent statements.

Statement 1

Among all the planar shapes with the same perimeter, the circle has the largest area.

Statement 2

Among all the planar shapes with the same area, the circle has the shortest perimeter.

Theorem

Statements 1 & 2 are the same.

Proof

As a shorthand, denote Statement 1 as A and Statement 2 as B. Assume A holds and let us prove B. Supposed, on the contrary, that B is false. Then for a given circle C there exists a figure F with the same area but with a perimeter shorter than C. Shrink C into a smaller circle C' whose perimeter equals that of F. The area of C' will clearly be smaller than that of C and consequently, it will be smaller than the area of F. Now, this contradicts our assumption that A holds: C' and F have the same perimeter but the area of the circle C' is less than the area of F. Thus A=> B.

The implication B=>A is proven in precisely the same fashion.

Note that neither A or B was proven but rather, that they are either both true or both false. Mathematical theorems in general have premises and conclusions and assert that the latter follow from the former. Human reasoning often resorts to the well known modus pones syllogism: if A=>B and A is true, then so is B. Oftentimes, the premise A seems to be so obviously true that a proof of A=>B is mistakenly substituted for a proof of B.

Coincidence?

In a group of 23 people, at least 2 people have the same birthday with the probability higher than 50%.

This could be reformulated the following way. There is a bag of 365 different items. We pick an item at random. Note which one it was and return it back into the bag. The question: how many items should we pick so that with a probability higher than 1/2 we would have picked the same item twice. The items in the bag are assumed to be thoroughly mixed before every trial. Is it not surprising that all it takes is 23 items?

To prove our assertion let us start with just two people. What is the probability p₂ that two random persons have the same birthday? It's easier to answer a related question. What is the probability q₂ that two random persons have different birthdays? Obviously, p₂ + q₂ = 1. Thus answering one question we automatically get an answer to another.

Thus we have a person with a birthday which falls onto one of the 365 days and ask what is the probability that another person has a different birthday. Since any day out of 365 but the birthday of the first person would make a different birthday, q₂ = 364/365.

Consider now three people. What is the probability q₃ that no two of them have the same birthday? Obviously p₃ + q₃ = 1, where p₃ is the probability that at least two of the group have the same birthday. As before, take one fellow and his birthday. The second has now 364 days to choose from and, if the third was born on any of the remaining 363 days they would form a "no-overlapping-birthday" group. Thus q₃ = 364/365·363/365.

Proceeding in this fashion we'll get q₄ = (364/365)·(363/365)·(362/365), and so on. Since every fraction in these products is less than 1, the sequence q₂, q₃, q₄, ... is decreasing. Therefore, the sequence p₂, p₃, p₄, ... is increasing. Now, perhaps, it would be less surprising to learn that p₂₃ > 1/2. Recollect that

q23 = 364/365 · 363/365 · ... · 343/365

The Shapes Of Constant Width

Yes - there are shapes of constant width other than the circle.

BUT

No - you can't drill square holes. However, saying this was not just an attention catcher. The truth is, you can actually drill holes that are almost square - drilled holes whose border includes straight line segments!

Now then let us define the subject of our discussion. First we need a notion of width. Let there be a bounded shape. Pick two parallel lines so that the shape lies between the two. Move each line towards the shape all the while keeping it parallel to its original direction. After both lines touches the figure, measure the distance between the two of it. This will be called the width of the shape in the direction of the two lines. A shape is of constant width if its (directional) width does not depend on the direction. This unique number is called the width of the figure. For the circle, the width and the diameter coincide.

The curvilinear triangle is built the following way. Start with an equilateral triangle. Draw three arcs with radius equal to the side of the triangle and each centered at one of the vertices. The figure is known as the Reuleaux triangle. Convince yourself that the construction is indeed results in a figure of constant width. Starting with this, we can create more. Rotating Reuleaux's triangle covers most of the area of the enclosing square. For the width=1 the following formula is cited in Eric's Treasure Trove of Mathematics.

which looks pretty close to 1, the area of the square.

Extend sides of the triangle the same distance beyond its vertices. This will create three 60^o angles external to the triangle. In each of these angles draw an arc with the center at the nearest vertex. All three arcs should be drawn with the same radius. Connect these arcs with each other with circular arcs centered again at the vertices (but now the distant ones) of the triangle.

There are many other shapes of constant width. May you think of any? There are in fact curves of constant width that include no circular arcs however small.

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Möbius Strip

Sphere has two sides.

A bug may be trapped inside a spherical shape or crawl freely on its visible surface. A thin sheet of paper lying on a desk also has two sides. Pages in a book are usually numbered two per a sheet of paper. The first one-sided surface was discovered by A. F. Möbius (1790-1868) and bears his name: Möbius strip. Sometimes it's alternatively called a Möbius band. (In truth, the surface was described independently and earlier by two months by another German mathematician J. B. Listing.) The strip was immortalized by M. C. Escher (1898-1972).

To obtain a Möbius strip, start with a strip of paper. Twist one end 180^o (half turn) and glue the ends together. For comparison, if you glue the ends without twisting the result would look like a cylinder or a ring depending on the width of the strip. Try cutting the strip along the middle line. People unacquainted with Topology seldom guess correctly what would be the result. It's also interesting cutting the strip 1/3 of the way to one edge. Try it.

Now once you know the trick, surely you would like to find other one-sided surfaces. Before gluing the ends together you can twist the strip twice or even three times. Do you get one-sided or two-sided surface?

Acknowledgments:

http://www.mathsoft.com

http://www.cut-the-knot.org/do_you_know/moebius.shtml