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Chapter 1: Problem 169

Suppose we are given an \(m\)-gon (polygon with \(m\) sides, and including the interior for our purposes) and an \(n\)-gon in the plane. Consider their intersection; assume this intersection is itself a polygon (other possibilities would include the intersection being empty or consisting of a line segment). a. If the \(m\)-gon and the \(n\)-gon are convex, what is the maximal number of sides their intersection can have? b. Is the result from (a) still correct if only one of the polygons is assumed to be convex? (Note: A subset of the plane is convex if for every two points of the subset, every point of the line segment between them is also in the subset. In particular, a polygon is convex if each of its interior angles is less than \(\left.180^{\circ}.\right)\)

Short Answer

Expert verified
a) The maximal number of sides is mn.b) No, the result does not hold if only one polygon is convex.

Step by step solution

01

Understanding Polygon Intersection

Consider two convex polygons, an m-gon and an n-gon. If their intersection is also a polygon, we need to determine the maximum number of sides this intersection can have. Convex polygons ensure that the intersection is also convex and forms a polygon.
02

Use Euler's Formula for Convex Polygons

When two convex polygons intersect, the resulting intersection polygon can be considered in terms of Euler's characteristics. Euler's formula states that the intersection of two convex polygons is a convex polygon with vertices formed by intersections of the sides of the originating polygons.
03

Determine Maximum Number of Vertices

Consider that each side of the m-gon can intersect each side of the n-gon at most once. Thus, the maximum number of vertices for the intersection polygon will be the product of the numbers of sides, yielding mn vertices.
04

Maximal Number of Sides in Convex Intersection

The maximum number of sides of the resulting intersection polygon will equal the maximum number of vertices since each vertex corresponds to a side. Therefore, the maximum number of sides in the intersection polygon is mn for two convex polygons.
05

Consideration if Only One Polygon is Convex

If only one of the polygons is convex and the other is not, the intersection is not necessarily a convex polygon. It may have a more complex structure, potentially allowing for a greater number of sides. Therefore, the result in part (a) does not hold.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convex Polygons
Convex polygons are a special type of polygon where every interior angle is less than 180 degrees. This ensures no indentations in the shape.
Imagine a rubber band stretched around the shape, it will touch all sides exactly. This is a good way to visualize convexity.
Convex polygons have these properties:
  • Connecting any two points within the polygon always results in a line segment that stays entirely inside the polygon.
  • The intersection of two convex polygons always results in another convex polygon.
  • They are simpler to work with in mathematical problems.
Understanding convex polygons helps in solving intersection problems because they always produce predictable and manageable shapes.
Euler's Formula
Euler's Formula is a fundamental equation in Geometry that relates the number of vertices \(V\), edges \(E\), and faces \(F\) of a convex polyhedron. The formula is expressed as: \[ V - E + F = 2 \] Although this formula primarily applies to 3D shapes, concepts from it help in understanding polygons in 2D.
For convex polygons, this simplifies our work:
  • Each vertex in the intersection corresponds to an edge.
  • The number of edges equals the number of vertices.
When two convex polygons intersect, the intersection's vertices stem from the sides' intersections. Each intersection creates a vertex and an edge, matching Euler鈥檚 principles.
Maximal Number of Sides
To determine the maximal number of sides, we look at intersections:
  • Each side of an \(m\)-gon can intersect with each side of an \(n\)-gon at most once.
  • This means the maximal number of intersections (and thus vertices) is \(m \times n\).
  • Since each vertex corresponds to an edge, the intersection polygon has up to \(m \times n\) sides.
If at least one polygon is convex, we still follow this rule for one convex polygon intersecting a non-convex one. However, if neither is convex, complexities increase and the rule does not hold.
Understanding the maximal number of sides ensures accurate geometric predictions and clearer visualization of possible shapes.

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Most popular questions from this chapter

Define a function \(f\) by \(f(x)=x^{1 / x^{x^{1 / x^{x^{1 / x ^{. ^{. ^{. }}}}}}}} (x>0)\). That is to say, for a fixed \(x\), let $$ a_1=x, \quad a_2=x^{1 / x}, \quad a_3=x^{1 / x^x}=x^{1 / x^{a_1}}, \quad a_4=x^{1 / x^{x^{1 / x}}}=x^{1 / x^{a_2}}, \quad \ldots $$ and, in general, \(a_{n+2}=x^{1 / x^{a_n}}\), and take \(f(x)=\lim _{n \rightarrow \infty} a_n\). a. Assuming that this limit exists, let \(M\) be the maximum value of \(f\) as \(x\) ranges over all positive real numbers. Evaluate \(M^M\). b. Prove that \(f(x)\) is well defined; that is, that the limit exists.

Find the smallest possible \(n\) for which there exist integers \(x_1, x_2, \ldots, x_n\) such that each integer between 1000 and 2000 (inclusive) can be written as the sum, without repetition, of one or more of the integers \(x_1, x_2, \ldots, x_n\). (It is not required that all such sums lie between 1000 and 2000, just that any integer between 1000 and 2000 be such a sum.)

Define \(\left(x_n\right)_{n \geq 1}\) by \(x_1=1, x_{n+1}=\frac{1}{\sqrt{2}} \sqrt{1-\sqrt{1-x_n^2}}\). a. Show that \(\lim _{n \rightarrow \infty} x_n\) exists and find this limit. b. Show that there is a unique number \(A\) for which \(L=\lim _{n \rightarrow \infty} \frac{x_n}{A^n}\) exists as a finite nonzero number. Evaluate \(L\) for this value of \(A\).

Note that the integers \(a=1, b=5, c=7\) have the property that the square of \(b\) (namely, 25) is the average of the square of \(a\) and the square of \(c\) (1 and 49). Of course, from this one example we can get infinitely many examples by multiplying all three integers by the same factor. But if we don't allow this, will there still be infinitely many examples? That is, are there infinitely many triples \((a, b, c)\) such that the integers \(a, b, c\) have no common factors and the square of \(b\) is the average of the squares of \(a\) and \(c ?\)

For three points \(P, Q\), and \(R\) in \(\mathbb{R}^3\) (or, more generally, in \(\mathbb{R}^n\) ) we say that \(R\) is between \(P\) and \(Q\) if \(R\) is on the line segment connecting \(P\) and \(Q\) ( \(R=P\) and \(R=Q\) are allowed). A subset \(A\) of \(\mathbb{R}^3\) is called convex if for any two points \(P\) and \(Q\) in \(A\), every point \(R\) which is between \(P\) and \(Q\) is also in \(A\). For instance, an ellipsoid is convex, a banana is not. Now for the problem: Suppose \(A\) and \(B\) are convex subsets of \(\mathbb{R}^3\). Let \(C\) be the set of all points \(R\) for which there are points \(P\) in \(A\) and \(Q\) in \(B\) such that \(R\) lies between \(P\) and \(Q\). Does \(C\) have to be convex?
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