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Fun with Math

Sepp Buchmann

Simple Method to Calculate PI

Have you ever tried to calculate the value of PI? Here is a simple method that will demostrate the logic behind the PI. All you need is some basic eighth grade Pythagoras knowledge.

The Babylonians were well aware that six equilateral triangles (all three sides of equal length) could be arranged and formed into a hexagon, a quasi circle. How did early men know this?

They realized that seven round objects could be arranged into a bundle where six of the objects would exactly surround a center object without any gap. Mathematical minds would then realize that the distance from each object to the adjascent objects would in all cases be the diameter of the object. Linking up the centers of six surrounding objects created a six-sided quasi circle, and linking up the centers of the surrounding objects to the center object created six equilateral hexagons.

As the Babylonians used the ‘Base-60’ sexagesimal numbering system, the six sides of the hexagon was the origin of the number 360 ( 6 x 60) that is still used today for plane angles in geometry and trigonometry. The value of 60 arose from the way ancient people counted with the fingers. Touching the three joints of each finger with the thumb, one by one, accounted for 12, then multiplying the value 12 by each finger of the other hand: 5 x 12 = 60.

There are other possibilities how the Babylonians realized that six equilateral triangles can be formed into this perfect quasi circular shape. We can only speculate, but they may have discovered that the 'circular' shape actually represented a squashed cube, each of the 12 lines representing an edge of the cube, all of the same length. This can be illustrated by simply hiding three (3) of the lines; a perfect cube suddenly becomes visible.

For our project we start with the hexagon. Since all three sides of the triangles are equal in length, the sides of the triangle also represent the radius of our imaginary circle. If we add up the outer sides of all six triangles we get a value of six times the radius, or three times the diameter, close to the of 3.14 number for PI that we try to calculate. That is our starting position

To get to a refined, more accurate value of PI, we need to smoothen the circle. Let us take one of the six wedges. We cut the triangle in half and draw a separating line from the center of the circle up to the circumference, crossing the upper side of the triangle (R2). That full length of the line is equal to the radius, equal in length to each of the original triangle sides. Now, we need to calculate the value of the line section from the center point up to the crossing (red RA) and the line beyond the crossing (blue RB)). Here we need to do our first Pythagoras calculation. We know the length of R and we know the length of R2 (R2 is of course exactly half of the radius). RA is thus calculated as follows:

RA = Square Root of ((R*R) - (R2 * R2))

Now that we know the value of RA, we calculate the value of RB, by simply deducting the RA from the value of radius.

Now we are ready to refine the circle line further. The initial six sides will be doubled to 12. With our second Pythagoras calculation we obtain the hypotenuse of the yellow RB and R2 configuration, shown as a green line (S) in the image. The calculation is as follows:

S = Square Root of (( R2*R2) + (RB*RB))

With this calculation we now have 12 equal sections around our circle and the PI stands much improved at 3.13. But that is not good enough.

So we continue. Using the same steps as above, we cut the sections by half again, and again. The PI improves to 3.139, then 3.141. Once we get to 20 iterations, we have a perfect PI value of 3.14159265358976. Of course we write a simple program to do this (sample code below).

We can calculate to infinity; a number is never too small to be cut in half.

Sample code for the PI calculation in VB.NET:

Dim radius As Double = 1000
Dim sec As Double
Dim adjacent As Double
Dim opposite As Double = radius / 2
adjacent = Math.Sqrt((radius * radius) - (opposite * opposite))
sec = Math.Sqrt((opposite * opposite) + ((radius - adjacent) * (radius - adjacent)))

Dim iterations As Integer = 20
For i As Integer = 1 To iterations
adjacent = Math.Sqrt((radius * radius) - ((sec / 2) * (sec / 2)))
sec = Math.Sqrt(((sec / 2) * (sec / 2)) + ((radius - adjacent) * (radius - adjacent)))
Dim circumference As Double = sec * 6 * (2 ^ iterations)
Dim pi As Double = circumference / radius

Copyright 2009 Joseph Buchmann
You may use this text and the images with attribution to me and link to web page.

Pythagoras Theorem, Simple Proof Method

Here is a simple method to prove the Pythagoras Theory. All you need is a sheet of paper, a pencil and right-angle ruler, and a bit of eighth grade Pythagoras knowledge. It is that simple, I say it is almost simpler than pronouncing the name Pythagoras.

The object is to graphically demonstrate and prove that the Pythagoras Theorem gives the correct value of the third side of a right-angled triangle where we know the values of the other two sides. For our exercise we know the values of the width and height and we solve for the hypotenuse. In general, we want to prove the relationship of the three sides of any right-angled triangle.

Step 1: Draw a rectangle. Make it 3 Inches wide and 2 Inches high. This is the main shape, shown in the image as the rectangle with a thick border. We will refer to it below often as the ‘Rectangle’.

Step 2: Draw a diagonal line in the Rectangle from the lower left to the upper right.

Step 3: Now expand the diagonal and make it into a perfect square, with all sides having the same length. All corners must be at a 90 degree right angle.

Step 4: Draw lines from each corner in perfect horizontal or vertical directions, as marked with red arrows. Draw the lines in the order shown.

You have now four wedges, right-angled triangles, colored in yellow (A, B, C and D). Each triangle is half the size of the original Rectangle; the total area of all four rectangles thus equals twice the area of the Rectangle. In our example, each triangle is 3 Inches times 2 Inches with an area of 3 square Inches (3 x 2 / 2). Keep this number in mind for the proof calculation below.

You will notice a residual square between the triangles, shown with a cross-pattern. This perfect square is exactly an area equal to the squared value of the difference between the width and the height of the Rectangle. For example, if the dimensions of the Rectangle are 3 x 2 Inches, the difference between the width and the height is 1 Inch and the area of the residual square therefore is (3 - 2) x (3 - 2) = 1 square Inch.

Now, the area of the large square that you have drawn, with all four sides having the length of the diagonal of the Rectangle equals the total areas of the four yellow triangles plus the residual square, or twice the size of the Rectangle plus the residual square. You calculate the length of the side of the large square (being the diagonal of the Rectangle) is by simply calculating the square root of the large square.

Now here is the proof:

The diagonal line of your Rectangle is, using the traditional Pythagoras calculation:

Diagonal = Square Root of ((3 x 3) + 2 x 2)) = 3.6055

With a slightly modified but equivalent formula the calculation is:

Diagonal = Square Root of ((4 x Triangle Areas) + Residual Square Area) Thus; Diagonal = Square Root of ((4 x 3) + 1) = 3.6055

Presented visually, the following three images demonstrate how the four wedges plus the residual square together represent the area of the large square (the square value of the diagonal line). In the third image, the upper part represents the square value of Side A (large horizontal side) of the original rectangle; the lower part represents the square value of Side B (small vertical side). Together, the area is equal to the square value of the rectangle's diagonal line.

Click here for a printable puzzle sheet.

Click here for a printable copy of this page.

Copyright 2009 Joseph Buchmann
You may use this text and the images with attribution to me and link to web page.

Days of the Week

Have you ever wondered where the names of the week come from? You probably know that all the seven days are named after the wandering heavenly bodies. In the ancien world there were known just seven of these 'stars', each sacred and the embodiement of a God or Goddess. In English, sadly, only Sunday and Monday still bear the legacy name, all other days have been supplanted in time to pay homage to the Germanic pagan Gods related to these stars. In French, all the week days’ names still make reverence to the deities of the seven heavenly bodies, except Sunday.

Why are there seven days in a week? The seven-day period goes back to ancient times, probably originating from the 28-day lunar cycle, divided into the four lunar phases (new moon, increasing half moon, full moon, decreasing half moon). And the seven days tallied nicely with the seven heavenly bodies.

But have you ever thought and wondered why the days’ names are in the order that we are accustomed to? Why does Friday follow Thursday and not vice versa? The history of the origin of week day names is fascinating and it even includes some mathematics.

The wandering heavenly bodies, the planets, moon and sun, known to the ancient people are traditionally listed in the order of their orbits. They all have their own orbit, and historically the listing was in the order of the time length of their orbits.

Traditional order of the heavenly bodies known in antiquity, with orbits,
and French and English associated week day names:

1. Saturn (29 years) Samedi (Saturday)
2. Jupiter (12 years) Jeudi (Thursday)
3. Mars (687 days) Mardi ((Tuesday)
4. Sun (365 days) Dimanche (Sunday)
5. Venus (225 days) Vendredi (Friday)
6. Mercury (88 days) Mercredi (Wednesday)
7. Moon/Luna (28 days) Lundi (Monday)

But why are the days now out of order? The reason goes back to ancient Egyptian time. Even back in times of antiquity a day was split into 24 hours. That makes sense, 12 hours for the day and 12 hours for the night. In Egypt and in other ancient cultures the number 12 was a very important number, similar to our number 10 today. Why 12? Perhaps the twelve moons in a year, but more likely from finger joint counting, or one influencing the other. With one hand open, the ancient people used the thumb to count the joints of the four fingers; four fingers, each with three joints, twelve counts. The number 12 was also corraborated by the number of moons in a year This also explains the 'Base-60' sexagesimal numbering system used by the Babylonians and by us in our modern times when referring to minutes and seconds. A hand-full represents a value of 12, multiplying 12 by the number of fingers on the other hand, also used for counting, leads up to a number 60.

But it is very likely that the 12-hour day has a different origin, but one can argue that it is reconcilable with the joint-counting numbering method. As far back as 1600 BC well-to-do Egyptians owned portable time pieces. They were not sun dials as we know them now but cleverly designed shadow clocks that looked like 'modern razor blades', but much larger of course. The short-end of the 'L' shaped device, pointing up, had a cross-bar on top. This gadget was placed towards the sun, easterly in the morning and westerly in the afternoon. To control the level, a hanging perpendicular cord with a weight ('plumb line') was probably attached, and some seasonal adjustments were made. The cross-bar cast a shadow on the long horizontal bar; a long shadow when the sun was low and a short shadow when the sun was high. The horizontal bar was marked with five sections, each section representing an hour. The twilight hours, the early morning hour and the late afternoon hour, were not marked on the bar because the sun was down or not strong enough. So, five hours before noon and five hours after noon plus the two twilight hours added up to twelve hours.

But I digress. However, as you will see, the 24 hour cycle was instrumental in the development of the day names. Why are the names of the week in an order that is different from the traditional lineup of the stars. For simplicity I will call these heavenly bodies stars, which they are not really, other than the sun. As I mentioned, the order in the traditional lineup of stars was based on their length of orbit, 28 days for the moon and 29 years for Saturn, for example It is all simple and logical in a way.

In ancient Egypt, each hour of the day and night had a name, just as we have names for the days today. The name of the hour was in dedication to one of the seven sacret stars, the shining objects in the night sky, embodiments of the feared or benign Gods and Goddesses of the people. The names of the hours were in the strict order of the stars. There were only seven such stars but 24 hours, so every seven hours the cycle and names repeated themselves. The star God of the first hour after midnight was all-important; that God or Goddess controlled and owned the day, and the full day was named after him or her.

As the image shows, our demonstration starts at midnight with the all important and life giving Sun God. The first hour of the first day is thus dedicated to the Sun God and the whole day is named after him (Sunday), the second hour is dedicated to Venus, the third hour to Mercury, the fourth to the Moon, the fifth to Saturn, the sixth to Jupiter, the seventh to Mars (in the order of the traditional lineup), then back to the Sun. We continue though the 24 hours of the day giving each hour to its rightful God. On the first hour of the next day, continuing the uninterrupted sequence of stars, we arrive at the Moon God. So, the first hour of the second day is dedicated to the Moon God and the day is named after him (Monday). And so it continues. Each star God gets his turn and the day is nameed after the God of the first hour: Tuesday, Wednesday, and so on, 7/24. Use the image above and go thru the cycles (use your thumb and finger joints if you must…).

Or use a simpler method. Since the number 24 does not divide into 7, there is a remainder of 3, go thru the circle in jumps of three: 24,3,6,9,12,15,18. You reach the Stars in the new order, in the order of our week days.

It is absolutely amazing how you end up with today’s order of the week:

1. Sun Sunday Dimanche
2. Moon/Luna Monday Lundi
3. Mars Tuesday Mardi
4. Mercury Wednesday Mercredi
5. Jupiter Thursday Jeudi
6. Venus Friday Vendredi
7. Saturn Saturday Samedi

Whereas in French most weekdays still carry the name of a star, in English and other western languages many of the days took on names of Germanic pagan, like Tiw, Wotan, Thor, Frigg. There are some logical connections. Both the pagan Gods Tiw and the Mars are Gods of War. There are known associations between Gods Wodan and Mercury going back to Roman times. Thor and Jupiter are both Gods associated with Weather, and Friig is the Goddess of love and lust, in the image of Venus.

The Golden Ratio

The golden ratio is one of the most amazing phenomenon in nature. It is the ground where the universal law of nature is in harmony with mathematics. No wonder it is also called the divine proportion and golden number.

In mathematics and the arts, a rectangle has a golden ratio if the ratio of the longer side to its shorter side is the same as the ratio of the total of both sides is to the longer side. A rectangle in the golden ratio makes it the most aesthetic looking rectangle. Later in this article are described simple methods of calculating the golden ratio. You can even determine the golden ratio with a string.

In nature, generally, two related parts are either symmetrical or sized to relate each other in the golden ratio. A rectangle or ellipse is at its most aesthetic look and completeness if the width to height is in the golden ratio. Early artists and architects were well aware of the importance of the golden ratio. The Parthenon in Athens has many elements that follow the rules of the golden ratio. Most books printed in the 16th to the 18th century had height to width ratios exactly in harmony with the golden ratio, accurate to within millimeters. The divine proportion of the rectangles gives them the most appealing and pleasing appearance. Nature also uses the golden ratio. In the human body, for example, consider the sizes of the bones of the arm and hands: the upper arm, forearm, and the joints of the hand. Measuring these sections, the proportions of each joint to the next smaller joint amazingly are in the golden ratio. Almost everything in nature has some golden ratio aspects.

The Fibonacci Sequence

The numeric steps that lead to the golden ratio are known as the Fibonacci Sequence. The Fibonacci Sequence can be demonstrated by building a plane formation of squares. Start with one square, then add additional squares next to and equal in size to the widest side of the existing rectangle (or initial square). As more and more squares are added, the squares get larger and larger and the ratio between the short and the wide side of the resulting rectangle gets closer and closer to the Golden Ratio of 1.61803399…, but never exceeds it. The ratio is an irrational number, it is never exhausted. The sizes of the squares that you add represent the Fibonacci Sequence. The number in the Fibonacci Sequence is always the total of the previous two numbers, thus the sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. The eventual ratio of the square size to its immediate predecessor represents the Golden Ratio.

What is one of the most amazing revelations is that you see the Fibonacci numbers everywhere in nature. These numbers are Nature's way of counting. Consider the numbering of the growing and living things in nature: 5 teeth in children (each side), 8 teeth of adults, 5 fingers, 5 toes, petals in flowers, seeds on flower heads, and spirals in sunflowers, more often than not match Fibonacci numbers.

The FibonacciSquare

The value of the golden ratio is 1.618033… and is calculated by taking the square root of 5, adding 1 and dividing the result by 2. But you don’t need a calculator; you can obtain the golden ratio without calculation, simply by using a sheet of paper, a pen and a ruler as the demonstration below shows. You can also determine if a rectangle is golden with a simple pen and ruler exercise, or simply with a string or thread. Rendering the formula in a graphical presentation will make it easier for you to remember it.

How to determine if a rectangle is in the Golden Ratio?

Inside the large rectangle, bottom left, mark up a small rectangle. The width is the same as the height of the large rectangle; the height is half that size.

Draw a diagonal line inside the small rectangle, from one corner to the opposite corner.

Extend the diagonal line by the one-half of the large rectangle height. The line now represents the golden width. If the length of the extended line is equal to the width of the large rectangle, then the rectangle is in a golden ratio.

How to obtain the Golden Ratio of a rectangle?

1. Draw two squares, place them side by side, thus forming a rectangle.

2. Draw a diagonal line thru the rectangle, the "red line".

3. Extend the red line by the width of a square.

4. Cut the red line in half. The cut line represents the golden ratio width of the rectangle.

5. Draw a rectangle with the height of the earlier rectangle and a width being the length of the red line.

Copyright 2010 Joseph Buchmann
You may use this text and the images with attribution to me and link to web page.

The Names of the Months

The names given to the months have nothing in common with mathematics but one can argue that they have some relationship to numbers. It is an interesting subject; we give scarce thought to some weird annomalities rooted in our calendar. Do you know why in the world the residual leap-year days are adjusted in February instead of December; with the 30-day and 31-day months orderly alternating, why is there an exception for July and August; and why is September the nineth month when the number Seven is in its name, like septem in Latin? Just like the mystery of the sequence of week names that I explained in the section above, there are good explanations for all of these.

A month represents a moon cycle. The moon cycles don’t calibrate neatly into a ‘sun’ year. Twelve moon cycles represent about 254 days and a sun cycle is just over 365 days. Over the long span of history different approaches were used to handle the discrepancy.

In the early years of the Romans, the year started with March and consisted of only ten months. The two moons after the ten-months period were a nameless dead season in the midst of winter when nothing was growing. March, the first month of the year was named after Mars, the God of Wars, aptly named for the month when, after the quiet winter season, the soggy soil firmed up, daylight grew longer and young men were called to war to conquer new lands and bring home the booty. April has its origin in Aprilis, ‘the month that opens’, perhaps alluding to the buds open up in spring. It has been argued also that April may be named in honor of Aphrodite, the Goddess of Love and Beauty. With the young men far away fighting, leaving the wives home alone, perhaps a possibility. Nah. May and June were probably named after the Roman Goddesses Maiesta and Juno. The remaining six months were simply named after their respective numeric position (keep in mind that March was the first month): Quintilis (July) for the fifth month, Sextilis (August) for the sixth month, then Septilis, Octilis. Novem and Decern are the root words for the ninth and tenth months.

Around 700 BC the Romans decided to add the two ‘dead’ winter moon cycles at the beginning of the calendar year. Januarius was named after Janus the God of Gates, beginning and endings, his head appropriately depicted with two faces looking in opposite directions, looking back at the old year and looking forward to the new year. Februarius refers to Februa the festivals of purification celebrated in Rome that month. Adding the two months in front explains why the names of September to December no longer relate to their embedded number (Dec for 10 is now the 12th month).

The number of days in a given month varied. Originally the months had either 30 or 31 days. The Romans viewed odd numbers as more propitious and thus the 30-day months were changed to lucky 29-day months. The left-over days were handled differently over time but essentially taken care of in the month of February or with an extra month inserted after February. February, it appears, still has the standing as the ‘end month’ from the time before it was moved up front, still today being the handler of the residual days. This seems to give a hint that the two no-name winter months each had some previous individual identity since they were plugged in at the beginning of the year in their original position, January followed by February, why else would the second month be the residual month.

Arrive the Roman Caesars and Emperors with big egos. Julius Caesar wanted a month named after him; Quintilis the fifth month be named July, he ordered. Not to be outdone, Emperor Augustus claimed the sixth month Sextilis for him. Julius’ month had 31 days but Augustus’ Sextilis only 30 or 29. It was not acceptable for Augustus to have his month with a number of days lower and less favorable than that of his oncle Julius, so the number of days in August was increased to 31, and it still is today.

Ark of the Covenant

We found the Ark of the Covenant.

Not really, but it is interesting how Mathematics is so closely tied to shapes and forms; actually in reverse. The hexagon in particular is most revealing as the section on the Pythagorean Theorem (first article above) demonstrates. Take two perfect triangles and place them on top of each other in opposite directions, shifted one third down. This creates the image of the Star of David. Now stare at the hub of the Star of David and image three lines emanating from its center. There it is: A perfect cube, a floating box, the Ark of the Covenant.

Copyright 2009 Joseph Buchmann
You may use this text and the images with attribution to me and link to web page.