# Understanding Geometry – 2

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If you want to brush up your high school geometry knowledge, then KhanAcademy is a good place to start. For example, I learned a new proof of Pythagoras Theorem (there are 4 different proofs on KhanAcademy) which uses scissors-congruence:

In this post, I will share with you few theorems from L. I. Golovina and I. M. Yaglom’s “Induction in Geometry ”  which I learned while trying to prove Midpoint-Polygon Conjecture.

Theorem 1: The sum of interior angles of an n-gon is $2\pi (n-2)$.

Theorem 2: The number of ways in which a convex n-gon can be divided into triangles by non-intersecting diagonals is given by

$\displaystyle{\frac{2(2n-5)!}{(n-1)!(n-3)!}}$

Theorem 3: Given a $\triangle ABC$, with $n-1$ straight lines $CM_1, CM_2, \ldots CM_{n-1}$ drawn through its vertex $C$, cutting the triangle into $n$ smaller triangles $\triangle ACM_1, \triangle M_1CM_2, \ldots, \triangle M_{n-1}CB$. Denote by $r_1, r_2, \ldots r_n$ and $\rho_1, \rho_2, \ldots, \rho_n$ respectively the radii of the inscribed and circumscribed circles of these triangles (all the circumscribed circles are inscribed within the angle $C$ of the triangle) and let $r$ and $\rho$ be the radii of the inscribed and circumscribed circles (respectively) of the $\triangle ABC$ itself. Then

$\displaystyle{\frac{r_1}{\rho_1} \cdot\frac{r_2}{\rho_2} \cdots\frac{r_n}{\rho_n} =\frac{r}{\rho} }$

Theorem 4: Any convex n-gon which is not a parallelogram can be enclosed by a triangle whose sides lie along three sides of the given n-gon.

Theorem 5 (Levi’s Theorem): Any convex polygon which is not a parallelogram can be covered with three homothetic polygons smaller than the given one.

The above theorem gives a good idea of what “combinatorial geometry” is all about. In this subject, the method of mathematical induction is widely used for proving various theorems. Combinatorial geometry deals with problems, connected with finite configurations of points or figures. In these problems, values are estimated connected with configurations of figures (or points) which are optimal in some sense.

Theorem 6 (Newton’s Theorem): The midpoints of the diagonals of a quadrilateral circumscribed about a circle lie on one straight line passing through the centre of the circle.

Theorem 7 (Simson’s Theorem): Given a $\triangle ABC$ inscribed in the circle $S$ with an arbitrary point $P$ on this circle. Then then feet of the perpendiculars dropped from the point $P$ to the sides of the $\triangle ABC$ are collinear.

We can extend the above idea of Simson’s line to any n-gon inscribed in a circle.

Theorem 8: A 3-dimensional space is divided into $\frac{(n+1)(n^2-n+6)}{6}$ parts by $n$ planes, each three of which intersect and no four of which have a common point.

Theorem 9: Given $n$ spheres in 3-dimesnional space, each four of which intersect. Then all these spheres intersect, i. e. there exists a point belonging to all the spheres.

Theorem 10 (Young’s Theorem): Given $n$ points in the plane such that each pair of them are at a distance of at most 1 from each other. Then all these points can be enclosed in a circle of radius $1/\sqrt{3}$.

I won’t be discussing their proofs since the booklet containing the proofs and the detailed discussion is freely available at Mir Books.

Also, I would like to make a passing remark about the existence of a different kind of geometry system, called “finite geometry“. A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. You can learn more about it here: http://www.ams.org/samplings/feature-column/fcarc-finitegeometries

# Dimension clarification

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In several of my previous posts I have mentioned the word “dimension”. Recently I realized that dimension can be of two types, as pointed out by Bernhard Riemann in his famous lecture in 1854. Let me quote Donal O’Shea from pp. 99 of his book “The Poincaré Conjecture” :

Continuous spaces can have any dimension, and can even be infinite dimensional. One needs to distinguish between the notion of a space and a space with a geometry. The same space can have different geometries. A geometry is an additional structure on a space. Nowadays, we say that one must distinguish between topology and geometry.

[Here by the term “space(s)” the author means “topological space”]

In mathematics, the word “dimension” can have different meanings. But, broadly speaking, there are only three different ways of defining/thinking about “dimension”:

• Dimension of Vector Space: It’s the number of elements in basis of the vector space. This is the sense in which the term dimension is used in geometry (while doing calculus) and algebra. For example:
• A circle is a two dimensional object since we need a two dimensional vector space (aka coordinates) to write it. In general, this is how we define dimension for Euclidean space (which is an affine space, i.e. what is left of a vector space after you’ve forgotten which point is the origin).
• Dimension of a differentiable manifold is the dimension of its tangent vector space at any point.
• Dimension of a variety (an algebraic object) is the dimension of tangent vector space at any regular point. Krull dimension is remotely motivated by the idea of dimension of vector spaces.
• Dimension of Topological Space: It’s the smallest integer that is somehow related to open sets in the given topological space. In contrast to a basis of a vector space, a basis of topological space need not be maximal; indeed, the only maximal base is the topology itself. Moreover, dimension is this case can be defined using  “Lebesgue covering dimension” or in some nice cases using “Inductive dimension“.  This is the sense in which the term dimension is used in topology. For example:
• A circle is one dimensional object and a disc is two dimensional by topological definition of dimension.
• Two spaces are said to have same dimension if and only if there exists a continuous bijective map between them. Due to this, a curve and a plane have different dimension even though curves can fill space.  Space-filling curves are special cases of fractal constructions. No differentiable space-filling curve can exist. Roughly speaking, differentiability puts a bound on how fast the curve can turn.
• Fractal Dimension:  It’s a notion designed to study the complex sets/structures like fractals that allows notions of objects with dimensions other than integers. It’s definition lies in between of that of dimension of vector spaces and topological spaces. It can be defined in various similar ways. Most common way is to define it as “dimension of Hausdorff measure on a metric space” (measure theory enable us to integrate a function without worrying about  its smoothness and the defining property of fractals is that they are NOT smooth). This sense of dimension is used in very specific cases. For example:
• A curve with fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface.
• The fractal dimension of the Koch curve is $\frac{\ln 4}{\ln 3} \sim 1.26186$, but its topological dimension is 1 (just like the space-filling curves). The Koch curve is continuous everywhere but differentiable nowhere.
• The fractal dimension of space-filling curves is 2, but their topological dimension is 1. [source]
• A surface with fractal dimension of 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume.

This simple observation has very interesting consequences. For example,  consider the following statement from. pp. 167  of the book “The Poincaré Conjecture” by Donal O’Shea:

… there are infinitely many incompatible ways of doing calculus in four-space. This contrasts with every other dimension…

This leads to a natural question:

Why is it difficult to develop calculus for any $\mathbb{R}^n$ in general?

Actually, if we consider $\mathbb{R}^n$ as a vector space then developing calculus is not a big deal (as done in multivariable calculus).  But, if we consider $\mathbb{R}^n$ as a topological space then it becomes a challenging task due to the lack of required algebraic structure on the space. So, Donal O’Shea is actually pointing to the fact that doing calculus on differentiable manifolds in $\mathbb{R}^4$ is difficult. And this is because we are considering $\mathbb{R}^4$ as 4-dimensional topological space.

Now, I will end this post by pointing to the way in which definition of dimension should be seen in my older posts:

• Dimension ≡ Dimension of the underlying vector space
• Dimension ≡ Lebesgue covering dimension of the underlying topological space
• Special Numbers: update (note that here I am talking about “topological manifolds” of which “differentiable manifolds” are a special case)