Category Archives: geometry

Math is fun: website

https://colleenyoung.wordpress.com/2017/11/05/math-is-fun/

With thanks and regards to Colleen Young.

Pick’s theorem to pick your brains!!

Pick’s theorem:

Consider a square lattice of unit side. A simple polygon (with non-intersecting sides) of any shape is drawn with its vertices at the lattice points. The area of the polygon can be simply obtained as (B/2)+I-1 square units, where B is number of lattice points on the boundary, I is number of lattice points in the interior of the polygon. Prove this theorem!

Do you like this challenge?

Nalin Pithwa.

The Porter Lecture — Interview with Prof. Erik Demaine

Mobius and his band

There are some pieces of mathematical folklore that you really should be reminded about, even though they are “well-known” — just in case. An excellent example is the Mobius band.

Augustus Mobius was a German mathematician, born 1790, died 1868. He worked in several areas of mathematics, including geometry, complex analysis and number theory. He is famous for his curious surface, the Mobius band. You can make a Mobius band by taking a strip of paper, say 2 cm wide and 20 cm long, bending it around until the ends meet, then twisting one end through 180 degrees, and finally, gluing the ends together. For comparison, make a cylinder in the same way, omitting the twist.

The Mobius band is famous for one surprising feature: it has only one side. If an ant crawls around a cylindrical band, it can cover only half the surface — one side of the band. But, if an ant crawls around on the Mobius band, it can cover the entire surface. The Mobius band has only one side.

You can check these statements by painting  the band. You can paint the cylinder so that one side is red and the other is blue, and the two sides are completely distinct, even though they are separated by only the thickness of the paper. But,, if you start to paint the Mobius band red, and keep going until you run out of band to paint, the whole thing ends up red.

In retrospect, this is not such a surprise, because the 180  degrees twist connects each side of the original paper strip to the other. If you  don’t twist before gluing, the two sides stay separate. But, until Mobius (and a few others) thought this one up, mathematicians didn’t appreciate that there are two distinct kinds of surface: those with two sides and those with one side only. This turned out to be important in topology. And, it showed how careful you have to be about making “obvious” assumptions.

There are lots of Mobius band recreations. Below are three of them:

  • If you cut the cylindrical band along the middle with two scissors, it falls apart into two cylindrical bands. What happens if you try this with a Mobius band?
  • Repeat, but this time, make the cut about one-thirds of the way across the width of the band. Now, what happens to the cylinder and to the band?
  • Make a band like a Mobius band but with a 360 degrees twist. How many sides does it have? What happens if you cut it along the middle?

The Mobius band is also known as a Mobius strip, but this can lead to misunderstandings, as in aLimerick written by a science fiction author Cyril Kornbluth:

A burleycue dancer, a pip

Named Virginia, could peel in a zip,

But she read science fiction

and died of constriction

Attempting a Mobius strip.

A more politically correct Mobius limerick, which gives away one of the answers, is:

A mathematician confided,

That a Mobius strip  is one-sided,

You’ll get quite a laugh

if you cut it to half,

For it stays in one piece when divided.

Ref: Professor Stewart’s Cabinet of Mathematical Curiosities.

Note: There are lots of interesting properties of Mobius strip, which you can explore. There is a lot of recreational and pure mathematics literature on it. Kindly Google it. Perhaps, if explore well, you might discover your hidden talents for one of the richest areas of mathematics — topology. Topology is a foundation for Differential Geometry, which was used by Albert Einstein for his general theory of  relativity. Of course, there are other applications too…:-)

— Nalin Pithwa.

 

 

 

 

 

 

 

Geometry problems for Pre-RMO

Practice Quiz:

  1. Prove that the median of a triangle which lies between two of its unequal sides forms a greater angle with the smaller of those sides.
  2. Point A is given inside a triangle. Draw a line segment with end-points on the perimeter of the triangle so that the point divides the segment in half.
  3. If the sides of a triangle are longer than 1000 inches, can its area be less than one inch?

You are most welcome to share your answers,

Nalin Pithwa

Geometric maxima minima — a little question for IITJEE Mains

Problem:

Of all triangles with a given perimeter, find the one with maximum area.

Solution:

Consider an arbitrary triangle with side lengths a, b, c and perimeter 2s=a+b+c. By Heron’s formula, its area F is given by

F = \sqrt{s(s-a)(s-b)(s-c)}

Now, the arithmetic mean geometric mean inequality gives

\sqrt[3]{s(s-a)(s-b)(s-c)} \leq \frac{(s-a)+(s-b)+(s-c)}{3}=\frac{s}{3}

Therefore, F \leq \sqrt{s(\frac{s}{3})^{3}}=s^{2}\frac{\sqrt{3}}{9}

where inequality holds if and only if s-a=s-b=s-c, that is, when a=b=c.

Thus, the area of any triangle with perimeter 2s does not exceed \frac{s^{2}\sqrt{3}}{9} and is equal to \frac{s^{2}\sqrt{3}}{9} only for an equilateral triangle. QED.

More later,

Nalin Pithwa

PS: Ref: Geometric Problems on Maxima and Minima by Titu Andreescu et al.

 

A ode to Geometry

Euclid alone

Has looked on Beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.

— Edna St. Vincent Millay (1923)

Math from scratch

Math from scratch!

Math differs from all other pure sciences in the sense that it can be developed from *scratch*. In math jargon, it is called “to develop from first principles”. You might see such questions in Calculus and also Physics.

This is called Axiomatic-Deductive Logic and was first seen in the works of Euclid’s Elements (Plane Geometry) about 2500 years ago. The ability to think from “first principles” can be developed in high-school with proper understanding and practise of Euclid’s Geometry.

For example, as a child Albert Einstein was captivated to see a proof from *scratch* in Euclid’s Geometry that “the three medians of a triangle are concurrent”. (This, of course, does not need anyone to *verify* by drawing thousands of triangles and their mediansJ) That hooked the child Albert Einstein to Math, and later on to Physics.

An axiom is a statement which means “self-evident truth”. We accept axioms at face-value. So, there are axioms, definitions, propositions, lemmas, theorems and corollaries.

Euclid’s geometry rests on the following fundamental axioms:

1)      There can be one and only one straight line joining two given points.

2)      (a) If O is a point in a straight line AB, then a line OC, which turns about O from the position OA to the  position OB must pass through one position, and only one,, in which it is perpendicular to AB.

(b) All right angles are equal.

3) (a) If a point O moves from A to B along the straight line AB, it must pass through one         position in which it divides AB into two equal parts.

3)(b) If a line  OP, revolving about O, turns from OA to OB, it must pass through one position in which it divides the angle AOB into two equal parts.

4) Magnitudes which can be made to coincide with one another are equal.

5) Playfair’s Axiom: Through a given point, there can be only one straight line parallel to a given straight line.

Note that these are the only basic assumptions to be used in geometric constructions also with ruler and compass.

PS: article reblogged and slightly modified:

Reference: To start geometry from scratch, you can start working from the first page of a classic text ” A School Geometry” by Hall and Stevens, Metric Edition; For example, Amazon India link is:

https://www.amazon.in/School-Geometry-H-S-Hall/dp/9385923331/ref=sr_1_4?crid=6QK90ZAHHFAU&keywords=a+school+geometry+hall+and+stevens&qid=1561783333&s=books&sprefix=A+School%2Caps%2C253&sr=1-4

Or Infibeam: https://www.infibeam.com/Books/school-geometry-part-1-6-pb-hall-h-s/9788183552806.html

I would like to add a few more details as there are some students/readers who want to pursue this further. {By the way, I have used the above reference only. One more thing,…Dover publications still prints/publishes/sells the original volumes of Euclids books}:

\textbf{Hypothetical Constructions}

From the above axioms, it follows that we may suppose:

i) A straight line can be drawn perpendicular to a given straight line from any point in it.

ii) A finite straight line (that is, a segment) can be bisected.

iii) Any angle can be bisected by a line (we call such a line its angle bisector).

\textbf{Superposition and Equality}

AXIOM: Magnitudes which can be made to coincide with one another are equal.

This axiom implies that any line, angle, or figure may be taken up from its position, and without change in size or form, laid down upon a second line, angle, or figure, for the purpose of comparison, and ti states that two such magnitudes are equal when one can be exactly placed over the other without overlapping.

This process is called superposition, and the first magnitude is said to be applied to the other. (Note: this is the essence of “congruency” relation in geometry).

\textbf{Postulates}

In order to draw geometric figures, certain instruments are required. These are — a straight ruler, and a pair of compasses. The following postulates (or requests) claim the use of these instruments, and assume that with their help the processes mentioned below may be duly performed:

Let it be granted:

  1. That a straight line may be drawn from any one point to any other point.
  2.  That a finite (or terminated) straight line may be produced (that is, prolonged) to any length in that straight line.
  3. That a circle may be drawn with any point as centre and with a radius of any length.
  4. some notes: Postulate 3 above implies that we may adjust the compasses to the length of any straight line PQ, and with a radius of this length draw a circle with any point O as centre. That is to say, the compasses may be used to transfer distances from one part of a diagram to another.
  5. Hence, from AB, (a given terminated line), the greater of two straight lines, we may cut off a part equal to PQ the less. Because, if with centre A, and radius equal to PQ, we draw an arc of a circle cutting AB at X, it is obvious that AX is equal to PQ.

More later,

Regards,

Nalin Pithwa.