Real Mathematics – Geometry #17

Why do we have round wheels in cars and bicycles?

Square Wheels

Let’s construct a square wheel like the following and test if it can be used as an efficient wheel:

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After turning the squared-wheel 45 degrees the square will look like the one on the right:

At this position it is clear even to the naked eye that the height of the wheel is taller comparing to its original state. If we continue turning the wheel for another 45 degrees it goes back to its original position.

Squared-wheel has big disadvantages. A car or a bicycle with squared-wheels will be doomed as height of the vehicle constantly changes.

Triangle Wheels

Among the triangles equilateral is the best one for constructing a wheel as all of its sides have equal length:

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After turning the triangular-wheel 60 degrees to the left, it will have the same height:

Although let’s go 30 degrees back and examine the height of the wheel:

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Here height of the wheel is clearly taller than the height of the original state. This proves that it is not appropriate to construct a triangular-wheel on a vehicle. Otherwise you might have problems with your spine.

Power of the Circle

The reason why circle is the most powerful and convenient shape for our vehicles is that neither of its height nor width changes while rotating. This separates circle from all the other polygons and makes it the best shape for wheels.

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Nonetheless, one wonders if circle is the only possible shape for wheels.

Reuleaux Triangle

Leonardo da Vinci is one of those names that appear in your mind when someone mentions Renaissance. The relationship between this magnificent figure and Reuleaux triangle comes from a world map that was found inside his pupil Francesco Melzi’s notes:

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This is known as one of the very first maps that included America. It is believe that this 1514-dated drawing was due to Leonardo da Vinci. If this is true, then it is safe to assume that da Vinci was the first person ever who used the Reuleaux triangle.

It was Leonhard Euler whom discovered the shape and explained it mathematically almost 200 years after da Vinci’s map. You probably realized this from my articles: “It is either Euler or Gauss.”

How come it is called Reuleaux? Because a century after Euler, a German engineer named Franz Reuleaux discovered a machine using Reuleaux triangle. In 1861 Franz Reuleaux wrote a book that made him famous and today he is known as the father of kinematics.

How to Construct Reuleaux Triangle?

I will show you my favorite method for its construction using three identical circles. First draw a circle that has radius r:

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Then pick a point on that circle as a center and draw a second circle again with radius r:

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At last, pick one of the crossings of the circles as center and draw a third circle with radius r:

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The central area of these three circles is a Reuleaux triangle:

When a Reuleaux triangle is rotating, it will have the same height at all times just like the circle:

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A bike that has a Reuleaux-triangled wheel.

One wonders…

  1. Using Euclid’s tools (a compass and an unmarked ruler) draw an equilateral triangle.
  2. Try to construct Reuleaux triangle with on the equilateral triangle.
  3. Can you contruct Reuleaux polygon(s) other than the triangle?

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #16

Smell of a Cake

I smell something wonderful. I beam myself up to the kitchen to investigate the source of this smell. I find it: My mother’s chocolate-chipped cake. As I leaned towards the cake, someone grabs my arm: Mom caught me…

I use the emotional card. She doesn’t buy it anymore. My opponent is experienced; my opponent is winning the battle!

As I was thinking of giving up, she offers me a deal. If I can cut three equal pieces out of this cake, one of the pieces will be mine.

Mom’s conditions:

  • Only instrument of measurement allowed for the cut is a compass.
  • Goal is to cut three pieces that have the same area. Size of the pieces is up to my cutting skills.
  • While making the cut, small differences (as if one area is 3,04 and other is 3,09) will be ignored by the mother.
  • Most important condition: Pieces must be in the shape of a ring.
  • You only have one chance for cutting. There is no turning back after the knife touches the cake.

Art of Cutting Cakes

I tried to find a method on paper because I satisfy all the conditions for the cut.

First I drew a circle that has center O:

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Then I created a chord which is as long as the radius of the circle:

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I placed the chord on random places inside the circle and marked chord’s midpoints:

I chose any of those marked points and drew a new circle that has center O:

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Then I followed the same procedure inside the new circle:

And finally I did the same things for the third time:

Areas which I colored with pens are equal to each other:

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Radius of the biggest circle=5 cm.
Radius of the second circle=4,34 cm.
Radius of the third circle=3,56 cm.
Radius of the forth (smallest) circle=2,55 cm.

For those who wonder the areas, you could calculate and see the approximate results.

One wonders…

I found out that it is possible to cut equal areas that are ring-shaped with using only a compass as mother asks.

Now think: How long the chord should be in order to cut the biggest possible piece?

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #15

Drawing a square

I am dealing with geometry and I imagine that I am in ancient Greece again. Aegean sea is in front of me and I am sitting on a marble between two huge white columns while holding an unmarked ruler and a compass.

First I draw a circle that has center at A and has radius r:

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Then I draw the same circle but taking its center at B this time:

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I connect the points A and B with a straight line. Then I draw two perpendiculars from the endpoints of the line AB:

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I connect the point E to the point F and end up with the ABEF square which has side lengths r:

Biggest circle that can be drawn into ABEF will have diameter r and touch the square at exactly four points:

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Area

In order to find the area of a square one can take the square of one side that gives r2.

To find a circle’s area one should multiply the square of the radius with π. In our inscribed circle we calculate the area as πr2/4.

Ratio of these areas would give π/4.

Weight

Now let’s make an experiment. For that all you need is some kind of cardboard cut as a square and a precision scale. Using the scale find the weight of the square-shaped cardboard.

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Then draw the biggest possible circle inside this square. Cut that circle out and find its weight with the scale.

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Since we are using the same material ratio of the weights should be equal to the ratio of the areas. From here one can easily find an approximation for the number π:

0,76/0,97 = π/4

3,1340… = π

One of the main reasons why we only found an approximation is that the cardboard might not be homogeneous. In other words the cardboard might not have equal amount of material on every point of itself.

Another reason for finding an approximation is that I didn’t cut the square and the circle perfectly.

One wonders…

Draw a circle and then draw the biggest-possible square inside that circle. Find their areas and measure their weights. See if you found an approximation.

M. Serkan Kalaycıoğlu

Real Mathematics – Life vs. Maths #2

Matches

For thousands of years people tried find a precise value for the number π (3,1415192…). At first this special number was thought to be seen only when there is a circle around. Within time π started to appear in places where scientist didn’t expect it to be. One of them was an 18th century scientist Georges Buffon.

Buffon came up with a probability problem named “Buffon’s needle problem” in 1777 when he came across with the number π. As I didn’t possess that many needles, I modified the problem as “Serkan’s matches problem”.

Buffon’s Needle Problem: Take a piece of paper and draw perpendicular lines on it with specific amount of space between them. Buffon wondered if one can calculate the probability of a needle that will land on one of the lines.

To start Serkan’s matches problem you need at least 100 matches, a piece of empty paper, a ruler, pen/pencil and a calculator.

First of all, draw perpendicular lines with 2 matches-length spaces between them.

Then just throw the matches on the paper randomly.

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Start collecting the matches which land on a line. At last you should use your calculator to divide the total number of matches to the number of matches landed on a line.

In my experiment out of 100 matches, 32 of them landed on a line. That gave me 3,125 which is close to the magical number π.

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In fact, 100 matches are not enough for this experiment. In my second try 34 matches landed on one of the lines which gave 100/34=2,9411… Obviously this is not close to π. More matches we use, closer we will get to π.

In an experiment back in 1980 2000 needles were used to analyze Buffon’s needle problem. Result was 3,1430… which is seriously close to the number π.

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You could go to https://mste.illinois.edu/activity/buffon/ and use this simulator which uses 1000 needle. In my first try I got 3,1496… You should try and see the result yourself.

In the future I will be talking about why a needle (or a match) is connected to the number π.

One wonders…

Try to do your own experiment and repeat Buffon’s needle problem for five times. Take the arithmetic average of your solutions and see how close you are to π?

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #4

Geometry has a sacred book: Elements. Author of Elements, Euclid, used only these tools when he discovered his geometry:

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A compass and an unmarked ruler… He showed what can or can’t be done with them in geometry.

Dividing a Finite Straight Line into Two Equal Parts

We already know that one can draw a straight line segment between any two points. Let’s say we have a straight line between the points A and B. Euclid found an ingenious method to divide AB into two equal parts using his only two tools. Let’s assume that AB is 6 cm.

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This way we will know that Euclid’s method works only if we find two parts that are 3 cm long.

According to Euclid one should take point A and point B as the centers of two equal circles with radius AB:

Euclid says that one should define the intersection points as C and D:

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Then he suggests one to connect C to D:

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At this point Euclid talks about two outcomes:

  1. CD is perpendicular to AB.
  2. CD divides AB into two equal parts.

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As seen in the picture CD really divides AB into two equal parts. One can use this method and see that those two lines are perpendicular with a quadrant.

Dividing a Random Angle into Two Equal Parts

Obviously Euclid didn’t stop there and tried to figure out if it was possible to divide any given angle into two equal parts. Let’s consider straight lines AB and BC intersects and form a 90-degree angle:

Euclid says that one should take B (the intersection point) as center and draw a circle with random radius. This circle will intersect AB and BC at two points: D and E.

Now Euclid tells us to use our compass and draw two equal circles that have centers D and E:

As seen above, these circles intersect at two points. Let’s choose the point F and connect it with B:

Euclid claims that the straight line BF divides the angle into two equal parts:

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We can see with our quadrant angle is divided into two equal parts of 45 degrees.

One wonders…

Is it possible to use Euclid’s methods and divide any given straight line into three equal parts?

Try to answer the same thing for an angle.

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #3

“There is no royal road to geometry.”

From Euclid to the king who asked Euclid if there is an easier way to learn geometry.

Up until now I have mentioned Euclid and his book Elements a few times. This masterpiece is actually a collection of 13 books and was considered as the source of only known geometry for thousands of years. Historical figures including Newton, Leibniz, Omar Khayyam and many others learned mathematics through Euclid’s Elements.

First book of Elements starts with 23 seemingly obvious and simple definitions. I will mention some of them below.

Elements Book I

Definition 1: A point is that of which has no parts. (Zero dimensions)

Definition 2: A line is length without breadth. (One dimension)

Definition 3: The extremities of a line are points.

Definition 4: A straight line is any one which lies evenly with points itself.

Definition 8: A plane angle is the inclination of the lines to one another when two lines in a plane meet one another and are not lying in a straight-line.

Definition 15: A circle is a plane figure contained by a single line such that all of the straight-lines radiating towards from one point amongst those lying inside the figure are equal to one another.

After reading these definitions for the first time, a few question marks popped up in my head.

For instance the first definition suggests that a point has no dimensions. If that’s so, how can one show a point lying on a plane?

Is it even possible to show something that has no dimensions?!

Which of these two can suggest a point to us? Obviously their sizes don’t matter and neither of them is an illustration of an actual point.

In this context, second definition is not different from the first one: One can’t draw something that has no breadth.

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Eighth definition is about angles. In order to draw an illustration for a random angle one must know how to draw lines, straight lines and dots.

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I’ve just showed you that even basic geometrical shapes are impossible to demonstrate. We can only imagine them in our minds. This means that in a way architects are selling illusions.

It is being told that mathematics has abstract and tangible parts. Whenever a student is dealing with abstract mathematics, teacher ought to give tangible examples so that student can comprehend with the subject easily. Nevertheless, we are helpless even when we want to give a full tangible explanation to a simple thing like a straight line.

Magic inside the Elements

In the first proposition of the first book of Elements given a random straight line, Euclid is showing us how to draw an equilateral triangle from that line.

Just to remind you, Euclid only used an unmarked ruler and a compass in his methods. Stop here and try to think of a way to construct an equilateral triangle from a random straight line.

Euclid’s Method

  1. Assume that we have a finite straight line AB.
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  2. Take AB as radius and draw a circle that has center A.
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  3. Now take AB as radius and draw another circle that has center B this time.
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  4. These circles will intersect at two points. Call one of them C.
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  5. Connect A to C. One can easily see that AB and AC are radii; hence they are equal in length.
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  6. Then connect B to C. One can observe that BC and BA are radii; hence they are equal in length.
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  7. AB and AC, BA and BC are equal. Since AB and BA are the same straight line one can conclude that AB=AC=BC.
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  8. These three straight lines construct an equilateral triangle.
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One wonders…

These methods are taken from a book that was written around 2300-2400 years ago. What I find fascinating about mathematics is that we are not even capable of showing what a dot is, but we can also explore other planets using the power of the language of mathematics.

Now use Euclid’s materials (an unmarked ruler and a compass) and try to draw the twin of a given random straight line. Hint: Analyze the second proposition of the book I of Elements.

M. Serkan Kalaycıoğlu

Real Mathematics – Geometry #8

Right way to cut a round cake

Almost everyone thinks of the same shape when someone mentions “a slice of round cake”:

However, this is not the correct way to cut a round cake. A letter was published in Nature magazine in December 20, 1906. It was written by a famous British scientist Francis Galton. Galton claimed that traditional way of cutting a round cake was faulty as after the cut exposed surfaces of the cake would start becoming dry almost instantly.

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Francis Galton’s letter.

Therefore he claimed that he found a “scientific principle” to cut a round cake.

Scientific way of cutting a cake is shown below:

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First Blood: Imagine two lines that are both parallel to the diameter and only a short distance away from it. As Galton suggested, one should cut the cake through these imaginary lines. Hence, exposed surfaces are same and they could be brought together. In order to keep the cake at a stable one piece position, you could use a rubber band.

Second Cut: You can do that like the first cut, but perpendicular to those cuts. In the end you would end up with four pieces of cake. They can be stuck together with a rubber band again.

This is the “scientific” way to keep your cake fresh.

In case you’d like to try Galton’s method without using a cake, all you need is a circle drawn on a paper:

One wonders…

Assume that you are on a Sunday brunch with your friends and pancakes arrived to the table. You realized that the waiter brought one extra pancake and everyone in the table wants that freebee.

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You all agreed to play a game in order to decide who gets the pancake.

Game: Everyone has a pancake on their plate. Everyone will try to cut his/her pancake to most pieces with three straight cuts. (You are not allowed to move the pieces of the pancake.) Winner(s) will get the pancake.

M. Serkan Kalaycıoğlu