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Playground

Rapunzel

Dido's Problem

Pythagoras

Trigonometry

Smart Joe

Fuzzy Logic

Cryptography

Mathematicians

Amplitude:
the biggest value of a periodically changing value.

Angle: an angle measures a degree of rotation. If you turn an object by a full rotation it seems to be in the same position again. A right angle is a fourth of a full rotation.

Angles are measured either in degrees where a full rotation equals 360°; or in radiens where a full rotation equals 2π.

Atbash : a simple monoalphabetic substitution in which the last letter represents the first (Z for a), the second to the last the second (Y for b), and so on. The system of this cipher is visible in its name: translated in our alphabet, the Hebrew Atbash would be called Azby.

 a b c d e f g h i j k l m n o p q r s t u v w x y z Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

Axiom:  a statment of which we assume that it is true without having to prove it.

Base: one side of a triangle; usually the bottom side. The other sides are called legs.

Caesar Cipher: a substitution. Each letter of the alphabet is substituted with the letter following a fixed number of positions later. Caesar, who lived around the birth of Christ, always shifted the letters by three steps.

Ciphertext: the encoded message. It may be based on one ore several ciphertext alphabets. Often a key is used as a basis of encoding the plaintext into a ciphertext.

Circle: a set of the points which have the same distance from a given point (the circle's center). The distance is called radius.

Perimeter and area of a circle depend only upon the radius. For the formulas, take a look at the figure on the right.

For an algebraic description of the circle, look under Descartes.

Clock System:

a finite system. This means that there is a defined amount of elements. The most commenly used clock system is the twelve-hour clock. Once we pass 12, we start again at 1, not taking into account the previous twelve hours; we don't keep track of how many times the hand went around.

Congruent:  in geometry the word congruent is used to describe figures that are indentical in shape and size. In modular arithmetic two numbers are congruent in modulo m if they have the same remainder after being divided by m. For example in mod 4 we say that 5 is congruent to 1, since 5:4=1 with the remainder 1 and 1:4=0, also with the remainder 1. Or if you picture it on a four-hour clock, we end up at the same place when we go one step or five steps, starting at 0.

Converse of the Pythagorean theorem: if one side of a triangle squared equals the sum of the squares of the two other sides, the triangle is a right triangle. For applications, read more in the history of the Pythagorean theorem.

Coordinate system: gives us the possibility to express the position of each geometric point with a pair of numbers. To do this, we draw to axes. Their intersection is called origin. The horizontal line is often called x-axis and the values on it increase going to the right. The values on the vertical axis, the y-axis, increase going upwards. The two axes don't have to have the same scale. The origin's coordinates are defined as (0,0), that means that its x-value (abscissa) and its y-value (ordinate) are 0.

On the left of the origin, the x-coordinates are negativ. Below the origin, the y-coordinates are negativ.

This results in four different quadrants. In the first one, you have positiv values for the abscissa and the ordinate. In the second quandrant, only the ordinate is positiv, in the third they both are negativ, and in the fourth the abscissa is positiv and the ordinate negativ.

Try out the grid on the left. You can choose some points and look at their coordinates.

Fermat and Descartes developed the idea of the coordinate system, often called Cartesian coordinate system after the Latin version of Descarte's name des Cartesius. The main idea behind it is the reduction of geometrical shapes to the simplest shape in geometry, the line.

Cryptography: the studdy of encrypted messages. The Greek word "cryptos" means "hidden" and"graphein" means "to write". It is the meaning which is hidden, not the message itself. Read about hidden messages under Steganography.

Take a look at Rapunzel's homepage!

Decode: (or decrypt); to find the hidden meaning of an encrypted message. If the key is known, this is usually an easy job. Otherwise it can become very difficult or even impossible. The only cipher which is absolutely impossible to decode is the one-time-pad. There are other encrypting methods which are currently impossible to decode, but in contrary to the one-time-pad, they will be decodable as soon as computers exist which are fast enough.

Definition:  a statement about a symbolic abbreviation which tells us how the symbol is to be used.

Dido: founder of the city Carthage. She emigrated from Phoenicia around 800BC. Her story was retold again and again. 300BC, for example, the historian Timaeus of Turomenium wrote about her, 200AD Vergil adapted her story, in the Middle Ages Petrarch, Ovid, and Boccacio. Around 1800 Charlotte von Stein, known as Goethes friend, dramatized Dido's story. The newest and shortest version is published in Mathematica Ludibunda.

Drag: when using a mouse on a computer, press the left mouse button on top of the object you want to drag and move the mouse while the left mouse button is still pressed. (On macs just press the only button available.)

Element: an element is a member of one or more sets.

Equilateral Triangle: a triangle with three equal sides and three equal angles. The Pythagorean theorem helps us to find out the height, so we only have to know the lenght of the side (a) to calculate the area.

ha = a√3/4

A = a × ha

A = a²/4√3

Finite System:  a mathematical system with a defined number of elements. For example the clock we use every day has exactly 12 characters (if you add 1 hour to 12 o'clock you get 1 o'clock again). Every whole number (0,1,2,3,...n) is congruent to a member of the original set. Another example of a finite system is the alphabet with its finite number of letters.

Four-hour System:a mathematical system based on a clock for four hours, numbered 0,1,2, and 3. This system can also be written as mod 4 (modulus: latin for measure; modulo 4: measuring in groups of four). In mod 4, two numbers are congruent if they differ by a multiple of four.

Frequency: number of periods per time. The number of periods per second is measured in hertz (hz for short). We have a frequency of 1hz if a wave has exactly one period per second. If we have 5 periods per second, we have a frequency of 5hz.

Function: a special type of relation. If we have two sets A and B, then there is only one corresponding element in set A for each element of set B. For the elements of set B there may be more than one corresponding element in set A.

Let's look at an easy example. If we have brother and sister, and the brother is three years older than his sister, we can write the relation between their ages like this:
y = x + 3, if the brothers age is y and the sisters age is x.

This is the formula which allows us to find the value y for each value of the variable x.

How the children's ages depend on each other can be written as a function like this:

f(x) = x + 3

That means that the value f(x) depends on the value of x. If x is 15, for example, then the value f(15) is 15 + 3 = 18. So if the sister is 15, her brother is 18.

We can show this relationship in a table. On the left you see the values of x, on the right the corresponding values of y.

 x y 1 4 2 5 3 6 ... ... 49 52 ... ...

Another way to show the relationship between x and y is graphing. We draw a coordinate system, take the sister's age as x-axis and her brother's age as y-axis. Now we put the values of the table above into the coordinate system and get a graph of the function
f(x) = x + 3.

As you can see, the graph of this funcion is a line. Graphs of functions may have other shapes, of course. By graphing trigonometric functions, for example, you get curves. In fuzzy Logic, working with graphs is very important, too.

Fuzzy Set: a group of elements which satisfy a certain criterion to some degree. For example, if you have the set of red apples, the set consist of all the apples which are at least a little bit red. The completely red apples are 100% members of this set. The apples which are red, but also have other colors, are members of the set to a certain degree. A completly green apple is not a member of this set; it does not have any red in it so it doesn't satisfy the criterion.

Height:

in a triangle the perpendicular from a point (angle) to the opposite side (or its extension).

Hertz: unit for the frequency; tells us how many periods there are in 1 second.

Hypotenuse: in a right triangle the leg opposite the right angle; the longest of the three legs.

Infinite System:  a mathematical system with an undefined number of elements. For example, the set of whole numbers goes on forever (you can always add 1 to get a bigger number).

Inversion: we get the inversion of a ratio if we change the order of the two numbers. The inversion of 1/5 is 5/1.

There are also functions which have an inversion, addition for example is the inversion of substraction, multiplication the inversion of division. In modular arithmetic, however, these functions are not inversions of each other.

The tangent is the inversion of the cotangent which means tanA=1/cotA; compare the graphs of these two functions.

Isosceles Triangle: a triangle with two equal sides. The height hc is easily calculated with the help of the Pythagorean theorem, so we only need the sides a and c to calculate the area.

A = c × hc

c = √a² - (c/2)²

Key: a cipher can be combined with a key. If we choose the Caesar cipher, we can determine with a key how many positions the plaintext alphabet has to be moved to get the ciphertext alphabet. For example, if we take the key 12, the plaintext and ciphertext alphabets would look like this:

 a b c d e f g h i j k l m n o p q r s t u v w x y z M N O P Q R S T U V W X Y Z A B C D E F G H I J K L

If we use the Vignère Cipher, the key says which of the different ciphertext alphabets has to be used. For example, if we use "math" as our key, we use the "m" alphabet to enrcypt the first letter of our plaintext. Then we use "a" to encrypt the second one, "t" for the third one and "h" for the fourth one. Aterwards we start with the "m" again, etc.

 a b c d e f g h i j k l m n o p q r s t u v w x y z M N O P Q R S T U V W X Y Z A B C D E F G H I J K L A B C D E F G H I J K L M N O P Q R S T U V W X Y Z T U V W X Y Z A B C D E F G H I J K L M N O P Q R S H I J K L M N O P Q R S T U V W X Y Z A B C D E F G

Mathematica Ludibunda: Mathematica Ludibunda is Latin and means "playful math" or "math made easy". Actually, "mathematica" can't be translated only as "math". Not long ago, mathematics included many other now seperate sciences. Physics, astronomy, music, philosophy and even art can also be described with the term "mathematica". The Greek word "mathema" means science in general.

The book got the name Mathematica Ludibunda because it is not only fun to read (playful math) with its many magic pictures to spice up the texts, but it is also easy to understand even if you're not a math professor. And don't stop reading as soon as you see "math...". This book contains parts of just about every meaning mathematica can have.

Modular Arithmetic: a mathematical system with a finite set of numbers; modulo m has m different numbers (modulo 5 for example the numbers 0,1,2,3, and 4). If a number differs by multiples of m, it is said to be congruent. Let's take modulo 3 for example. The element of this set are the numbers 0,1,and 2. In this system 3 is congruent with 0; and 4 with 1; 5 with 2; 6 with 0 and so on.

Monoalphabetic Substitution: each letter of the alphabet gets substituted by another letter. This substitution remains constant over the whole message (mono=one), in contrary to the polyalphabetic substitution (poly=more than one).

Let's look at an example with the keyword "math".

Plaintext alphabet
a b c d e f g h i j k l m n o p q r s t u v w x y z

Ciphertext alphabet
M A T H B C D E F G I J K L N O P Q R S U V W X Y Z

Decode QMOULZBJ!

One-time-pad: a polyalphabetic substitution similar to the Vignère cipher. Instead of using the same key over and over again like the Vignère cipher, it uses a pad with a randomly picked string of letters which determine the ciphertext alphabet to be used. Of course only the sender and receiver are allowed to know the order of the letters. A one-time-pad might look something like this:

asucetkbxgxfproasoeu
gprcegytubjvqsuhetol
anemkbqvenouhegpnsav

A one-time-pad is completely undecodable if it is used only once. Since every letter has the possibility to become any other letter, it is easy to read anything you want to out of a message encoded with a one-time-pad. The only problem is that the sender has to give the receiver a one-time-pad for every message he sends. Even if the pad is only used twice, it makes it possible for experts to decode the message.

a statement which seems to contradict itself. Two famous examples are the Dragon's Paradox and the Russell's Paradox.

Period:
the lenghts between two points which are surrounded by the same pattern.

Perpendicular: a line which intersects with another line in a right angle. A perpendicular is the shortest line between a point and a line.

A triangle has three perpendiculars to the bisection point of the legs which meet in point. This point is center of a circle (called circumcircle) on which are the triangle's corners.

The heights are also perpendiculars. And they, too, meet in a point (called orthocenter).

Plaintext: the original message. The plaintext alphabet is our normal alphabet. It's usually written with lower case letters, the ciphertext alphabet, which is used to write the ciphertext, with capital letters.

Polyalphabetic Substitution: contrary to the monoalphabetic substitution , the substitution of the letters change with each letter. This means that the "a" in the plaintext can be different letters in the ciphertext. The key determines which ciphertext alphabets and in what order they get used.

Let's look at an example. If we chose the cipher and "math" as our key, the alphabets look like this:

 a b c d e f g h i j k l m n o p q r s t u v w x y z M N O P Q R S T U V W X Y Z A B C D E F G H I J K L A B C D E F G H I J K L M N O P Q R S T U V W X Y Z T U V W X Y Z A B C D E F G H I J K L M N O P Q R S H I J K L M N O P Q R S T U V W X Y Z A B C D E F G

Decode DAIBZZXS!

Polygon: a closed shape with straight lines as borders. A regular polygon has equal sides and angles. To calculate the area, we have to divide it into triangles. If we sum up all the sides of the polygon, we get the polygon's perimeter.

Pythagorean Number Triplets: three whole numbers a, b, and c, for which the following is true:
a² + b²= c².
The smalles possible triplet with the numbers 3, 4, and 5 was already known to the Egyptians. They used it to build right triangles (Converse of the Pythagorean theorem).

On this website you will find more about triplets: www.faust.fr.bw.schule.de/mhb/pythagoren.html

Ratio: relationship between two parts. An example is the relation between the lenghts of two sides. If we compare 10ft and 2 ft, the ratio is 5:1, since 10:2 equals 5:1. A ratio can be written as x:y or as x/y.

Rectangle: a polygon with four right angles and four sides. The opposite sides have the same lenghts.

The figure on the left shows you how to calculate the area A and the perimeter P.

A special rectangle is the square with its four equal sides.

Right Triangle: a triangle with a right angle (90°). The side opposite the right angle is called hypotenuse.

It's easy to calculate the area of a rigth triangle:
A = ½ ab

Right angle: an angle with one fourth of a full rotation.If you have a line and a point which is not on the line, the shortest path from the point to the line meets the line at a right angle.

Set: a group of elements which satisfy a certain criterion. For example, every apple is a member of the set of apples, since the criterion for the set of apples is to be an apple. A pear, on the other hand, is not a member of the set of apples; it is not an apple and therefore doesn't satisfy the criterion.
Another example for a set is the set of numbers used in the four-hour system. Its elements are 0,1,2,and 3.

Sound Waves:  waves which can travel through matter such as air or water. Sound waves are usually generated by some vibrating object. For example, if you hit a tuning fork, the tips of the fork will start to move back and forth in a pattern which can be described by a sine wave. As the tip moves, the air molecules which are in the way of the tip of the fork will be commpressed together; this increases the pressure. The molecules behind the tip, however, will get spaced farther apart; this decreases the pressure . The air tries to cancel out the difference in pressure -> the molecules which are in parts of high pressure will move to parts of low pressure. At some point, while the molecules are moving towards the point of lower pressure, the pressure will be equal everywhere, but the molecules will continue to move towards the same point until they stop, increasing the pressure in the place which had a low pressure just a moment ago. This process continues and creates circular waves of compression and decompression, just as a rock creates waves when it is dropped into a pool of water. This goes on until friction has taken up all the energy.

 Our ears have the job to pick up these compressions and decompressions and to send them to our brain. For this they have an inclosed inner part which is sealed from the outside by a flexible membrane. When the molecules outside the membrane get compressed, they push onto the membrane with a greater force than the molecules inside the membrane, causing the membrane to move. The membrane is connected to the nerves by a complicated lever system to amplify the movement. The nerves send this movement to the brain which processes this information in some way that we can hear sound.

When we listen to sound, we hear some properties of the sound wave: the pitch of a sound is depending on the frequency of the sound wave; the higher the frequency of the wave, the higher the pitch. Sound can be described by sine waves. If the waves are not regular (periodic), we hear them as noise. The volume of sound depends on the amplitude of the wave. By adding waves together we can produce different tone colors. Each sound has its characteristic shape.

The Pythagoreans observed that the pitch of a string changes to a harmonizing pitch when we press down the string in such a point that the two parts of the string stand in a whole number ratio. If we press down in the middle of the string, for example, the new sound will be the higher octave of the pitch of the whole string. Here some ratios and their corresponding intervals:
1:2 octave
2:3 fifth
3:4 fourth

That means that a sound wave which goes through 400 complete cycles in a second (=400hz) is the octave of a wave with 800 cycles.

The corresponding of simple ratios with harmonious sounds was a proof for the Pythagoreans that nature was built in accordance with mathematical principles and that whole numbers and their ratios show the order of nature.

In the 16th century, scientist began experimenting not only with different lenghts of strings, but also with strings of diffent thickness (the thicker the string, the lower the pitch) and tension (the higher the tension, the higher the the pitch). With this knowledge the first string instruments as we use them today were developed.

Square: a shape with four equal sides and four right angles. To calculate the area you multiply the length of the side by itself. If you sum up the lenghts of the four sides, you get the square's perimeter.

Steganography: the studdy of hidden messages. The Greek word "steganos" means "covered" and "graphein" means "to write". One way to hide a message is to write it with the sap of a plant which becomes only visible when heated. In contrary to cryptographic messages, the meaning itself is not hidden. If someone finds the message, it is easy to read. Of course steganography and cryptography can also be used together.

Subset: a set which is an element of a different set. For example, the set of apples is a subset of the set of fruits. The set of fruits again is the subset of the set of food.

Substitution: each letter is exchanged for a different one. This can be done randomly or using a system. A famous example is the Caesar Cipher.

Transposition: the letters of a message are rearranged. This kind of encoding is known to have been used since the 5th century BC.

Here is an example: The letters of a message are rearranged in a way that every second letter is written under the previous letter.

r p n e s y h l o
a u z l a s e l

Afterwards you put the letters of the second row after the letters of the first row.

RPNESYHLOAUZLASEL

Triangle: a polygon with three sides. The sum of its angles equals 180°.

With one known side and the corresponding height, you find the area with the following formula:

A = ½ × c × hc

if you know for example c and its height hc.

You don't need an angle if all three sides are known.

A = √s(s-a)(s-b)(s-c), when s = ½(a+b+c)

Trigonometry gives us a lot more possibilities to calculate the area. For example, if you know side a and the angles, the following formula may help you: A = a²sinβsinγ ÷ 2sinα

Or, with two known sides and the angle between them:

F = ½absinγ

There are some special triangles for which we have easier ways to calculate the area: the isosceles triangle, the equilateral triangle, and the right triangle.

Trigonometry: (tri =three   gon=side   metrein=measure)  a mathematical science which studies the relationship between the sides and angles of triangles.

Vignère cipher: a polyalphabetic substitution. We can have up to 26 ciphertext alphabets for a plaintext. The alphabets are shifted exactly like the Caesar cipher. For every letter in the plaintext another ciphertext alphabet gets used. The order is determined by the key.

Blaise de Vignère lived in the 16th century. He was a French diplomat. He combined several ciphers to the one named after him.