MATHS IS EVERYWHERECOMENIUS PROJECT BOOK

www.mathsew.com

2011-2013

CHAPTER 1

SYMMETRY

Symmetry is everywhere you look in nature .

Symmetry is when a figure has two sides that are mirror images of one another. It would then be possible to draw a line through a picture of the object and along either side the image would look exactly the same. This line would be called a line of symmetry.

There are two kinds of symmetry.

One is bilateral symmetry in which an object has two sides that are mirror images of each other. The human body would be an excellent example of a living being that has bilateral symmetry.

The other kind of symmetry is radial symmetry. This is where there is a center point and numerous lines of symmetry could be drawn.

The most obvious geometric example would be a circle.

SHAPESSphere:

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

The shape of the Earth is very close to that of an oblate spheroid, a sphere flattened along the axis from pole to pole such that there is a bulge around the equator.

Hexagons:

Hexagons are six-sided polygons, closed, 2-dimensional, many-sided figures with straight edges. For a beehive, close packing is important to maximise the use of space. Hexagons fit most closely together without any gaps; so hexagonal wax cells are what bees create to store their eggs and larvae.

Cones:

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex.

Volcanoes form cones, the steepness and height of which depends on the runniness (viscosity) of the lava. Fast, runny lava forms flatter cones; thick, viscous lava forms steep-sided cones.

Fibonacci Spiral:

If you construct a series of squares with lengths equal to the Fibonacci numbers (1,1,2,3,5, etc) and trace a line through the diagonals of each square, it forms a Fibonacci spiral.

Many examples of the Fibonacci spiral can be seen in nature, including in the chambers of a nautilus shell.

SymmetrySymmetry has two

meanings.The first is a vague sense of harmonious and beautiful proportion and balance.

The second is an exact mathematical "patterned self-similarity" that can be demonstrated with the rules of a formal system, such as geometry or physics.

In Mathematics and many of applications thereof, there is great importance placed on the idea of symmetry and the methods of transforming one thing into another.

Forms of Mathematical

Symmetry

Mathematical symmetry, according to the dictionary, is "a geometrical or other regularity that is possessed by a mathematical object and is characterized by the operations that leave the object invariant." The word itself is derived from the Latin symmetria, which simply refers to an "agreement in dimension, proportion and arrangement

But what does this mean, exactly?Human beings (and other animals) are certainly familiar with a great many examples of symmetry in everyday life. This concept absolutely abounds in nature: The human body certainly posses a certain degree of superficial symmetry - cut it down the middle and the two halves will basically be identical (but opposite).

Likewise, there is a symmetry of many internal organs (two kidneys, two lungs, etc.) and of other natural phenomena, such as plants and even geological formations. Symmetry is everywhere.

What does this have to do with mathematics?Quantifying of SymmetryMathematicians have found many wonderful uses for the idea of symmetry, beginning as far back as the early Greek and Alexandrian geometers such as Euclid and Pythagoras, who were able to make ample practical use of the symmetry of certain shapes and formations in developing their clever proofs.Today, the mathematical idea of symmetry has grown to the point where one can point so several very distinct "forms" of symmetry, each of which hold their own value within the world of mathematics.

First, there is what is called reflectional symmetry. This might also be referred to as "mirror-image" symmetry, as it merely means that a certain object (or function, in mathematics) will be identical if viewed in respect to its mirror image. As an example of this, one might look to the function f(x)=x2, which any student of mathematics will easily recognize as a simple parabola which, when graphed, possesses reflectional symmetry along the x-axis of a Cartesian coordinate system.

Rotational symmetry, on the other hand, means that a mathematical expression is equal to another mathematical expression after having been "rotated" around a certain point by a certain degree. One example of rotational symmetry might be found in a function which is known as an "odd" function, which refers to a function with 180-degree rotational symmetry, such as: f(x)=x (which translates to a line passing through the point (0,0) at a forty-five degree angle. When one half of this line is rotated around the origin by 180 degrees, it will equal the other half. This is rotational symmetry.

Translational Symmetry is somewhat less well-known than these first two, and refers to two expressions which remain identical after undergoing any number of "translations." For instance, if one begins with two symmetrical expressions, x and y, these could be considered to be translationally symmetrical if they remained in the same relation to one another after each undergoing a certain translation (for instance, if one was to multiply both expressions by 2, or take them to a certain power).

For translational invariant functions it is .

Symmetry, geometry … dance?

Dance comes in many different forms. Professional dancers can work alone or in a group. Singers and musicians often use dance routines as part of their performances, and people dance to exercise and have fun. Maybe you even take dance lessons at school or in your spare time.

What you may not realise is that all types of dancing, from break-dancing to ballet, use ideas found in maths. Dancers must understand symmetry and geometry, as well as being able to count in time to the music. Choreographers also use these ideas to design new dances.

In a spinThink about a dancer spinning around. To stay in control and not get dizzy, dancers use a technique called ‘spotting’. As they turn their body, they keep their head fixed for as long as possible, and then quickly rotate their neck to catch up with their body.

They try to keep their head looking in the same direction after each rotation, because this helps them to balance and prevent dizziness.

Why can’t dancers just spin around continually? It is easy to keep a ball spinning because a sphere has an infinite or unlimited number of rotational symmetries. The human body has no rotational symmetry, meaning anyone trying to spin continually will soon lose their balance and fall over!

The power of symmetry

We may not have rotational symmetry, but we do have another type of symmetry called mirror symmetry. You may have also heard this called line or reflection symmetry. Imagine a line going from the top of your head to the ground between your feet. The left side of your body looks like a mirror image of the right side. You might have a few freckles in the wrong place, but your left arm matches your right arm, your left leg matches your right leg, and so on. We call this imaginary line the line of symmetry.

Symmetry is important, but so is geometry. Dancers form shapes with their bodies, and choreographers think about how to use lines and angles to make their dances more interesting. A choreographer called Rudolf Laban, has even created a system of notation for dance that can be manipulated like a mathematical equation.

CHAPTER 2

All the science requres mathematics (…)

Who is ignorant of mathematics cannot know the

other sciences nor the affairs of this world’

Roger Bacon

The main goal of the school mathematical-social club was looking for the maths signs in the

daily life. The activities of this club focused on the different tasks. The team prepared:

the school magazine the trip around our region the photo competition the exhibition of students’ photos

1. ‘Our passions in Maths. Maths is everywhere’- it’s a tittle of the school magazine, where the students could present their opinions and describe their passions. Turn out that their hobbies are concerned with maths and their proved it. Maths is everywhere. When we doing shopping, making a cake, building house or doing the chemical experiences.

Wiktoria and Karolina pointed that their favourite activities-a horseback riding, is also connected with maths. In show jumping you use Pythagoras theorem to find the most effective take off point. To do this the angle must be 45 degrees, and the distance of take off should be the height of the fence. In sporting you can do things like distance between posts, the degrees the horse bends to get around them and the distance to keep from the poles. In racing you can do stuff about betting, area of the track, speed of the horse, average stride in between the posts of 100m.

2. Maths in architecture

Math is used architecture every angle, wall legnth and everything like that must be exact so the building does not fall over when a lot of people go inside of it. Mathematics is essential to the study and practice of architecture. Every detail, big nor small should have exact measurements. The angles of the roof, the thickness of a wall, the amount of materials that will be used, calculating the exact location of a building and even the number of detail, these all includes math.

A school trip

In April 2012 forty- four students and three teachers went on a school trip around our region. They were looking for the elements in architecture connected with Maths. They sightseeing some interesting places. First of all, the St. Martin Church in Opatów which was built in the 12 th century, based on the latin cross plan.

The excellent mathematical exemple of architecture is the Krzyżtopór Castle or rather the ruine of palazzo in fortezza at present. It means that it was both palace and fortress. The total size of the complex is 1.3 hectares; the length of perimeter walls is 700 metres ; the total area – 70 000 square metres.

Krzyżtopór Castle had: 365 windows - as many as days of the year 52 rooms - as many as weeks of the year 12 ballrooms -as many as months of the year

Polish students are listening to the guide in the Krzyżtopór castle.

The Bishops’ Palace in Kielce There is an Italian geometrical

garden

behind the palace

3. The exhibition of students’ photos.

The students took part in the school photo competition- “Maths is everywhere “.

The students’ pictures were presented on the school exhibition.

The road signs

The clock on the market in Daleszyce The Decalog in Kielce

CHAPTER 3

“Musical form is close to mathematics - not perhaps to mathematics itself,but certainly to something like mathematical thinking and relationship.” composer Igor Stravinsky

From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics. Early Indian and Chinese theorists show similar approaches: all sought to show that the mathematical laws of harmonics and rhythms were fundamental not only to our understanding of the world but to human well-being. Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.

Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of pulse repetitivity, accent, phrase and duration – music would be impossible. In Old English the word "rhyme", derived from "rhythm", became associated and confused with rim – "number" – and modern musical use of terms like meter and measure also reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time and periodicity that is fundamental to physics.

To this day mathematics has more to do with acoustics than with composition, and the use of mathematics in composition is historically limited to the simplest operations of counting and measuring. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.

Mathematics is strongly associated with music. one such example is the musical interval.In music theory, an interval is the difference between two pitches. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such

as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord. In Western music, the most commonly used intervals are those formed by pairs of notes of a diatonic scale. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Intervals can be arbitrarily small, and even imperceptible to the human ear.

In scientific terms, an interval is the ratio between two sonic frequencies. For example, any two notes an octave apart have a frequency ratio of 2. This means that successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. For this reason, intervals are often measured in cents, a unit derived from the logarithm of the frequency ratio.

In Western music theory, the most common naming scheme for intervals describes two properties of the interval: the quality (perfect, major, minor, augmented, diminished) and number (unison, second, third, etc.). For instance, minor third or perfect fifth. These names describe not only the difference in semitones between the upper and lower notes, but also how the interval is spelled.

The speed of music:

Another thing is the speed of music. Tempo is nothing more than the speed of the beat. F or many years, Italian words were used to indicate tempo such as largo (broad), lento (slow), adagio (at ease), andante (walking), moderato (moderate), allegro (fast), and presto (very fast). The problem is that these designations were open to personal interpretation, and were therefore sort of ambiguous. The common practice today is to use metronomic markings, or beats per minute. For example, there are 60 seconds in each minute; if the tempo was such that a beat equaled one second, and each quarter note got one beat, the tempo would be would be quarter note q = MM 60. Most musical compositions fall in the range of MM 60-80, which is about the speed of human heartbeats or moderate walking (836-837). Of course, the tempo can easily be twice that fast if the music is intended for dancing, especially the music of those younger folks that are still full of energy!

The different schools and in different historical periods be interpreted differently. More closely to determine the rate was made possible by the invention of the metronome. Rarely, however, given the close rate, but rather the frequency range, leaving the player to interpret or conductor.

In addition to verbal expressions (mostly Italian) rate can be specified more precisely (in absolute terms) by the number of metronome beats per minute, and the identity of the

corresponding impact metronome rhythm. By default, this unit is a quarter, but you can choose any other entity, if it is more natural for the song (eg, eight note, half note).Pace music have a certain number of beats per minute so. BPM. Here are examples of numerical temperature and the rate of progress

Meaning the number of metric unitsper minute (BPM) *

slow pace

largo 40 - 60larghetto 60 - 66

lento 60 - 66adagio 66 – 76

moderate pace

andante 76 - 108moderato 108 - 120allegretto 120 – 132

fast pace

allegro 120 - 168presto 168 - 200

prestissimo 200 – 208

Symmetry in Music

What mathematics can we find in music?Well in music we have symmetry as well as in mathematics. Let’s take a simple gamma range from Do to Si and make it reflect on the mirror.

It is visible that we get an entirely different note. When we sum up those two tunes, we get a polyphonic music. Also, if we want to make even more beautiful tune we need to lag behind one beat.

We'll show you an example. First of all, you will hear an original melody. And now we will play two tunes at once. (VIDEO “Symmetry in Music”)

Tomaš Kulakovski, 9th form

Mathematics in Music Lessons

Music and Maths seems to be very different subjects. Maths is an accurate science and music is art, but, no matter how strange that could be, we found that it has strong connections with each other. Maths teacher Laura and Music teacher Eugenijus during the integrated lesson revealed to us the link between those two subjects. When they introduced the topic we were confused a bit. What will we do and why do we need Maths in Music lesson? However, we were surprised a lot.

First of all, the teacher told us that all the notes can be expressed by numbers. Pythagoras, the great mathematician and philosopher, was the first to notice this. We revised the notes and found that one semibreveequals – 1, half - note in music is equal ½ in Maths, crotchet – ¼ (viena ketvirtoji matematikoje) ir t.t.

The first task we had to accomplish was Bethoven song „The Dog“. We were asked to write down the song using fractions. It was quite easy so we did this task quickly. We were good at it. It was interesting to watch how Music suddenly became Math.

However, the second task was not so easy. Maths teacher asked us to do vice versa and turn 5 fractions to notes and add lacking notes because in our task we got three-quarter meter, so tact had to consist out of three-quarters. Just several students were able to do the task. O thers were not so smart or they needed more time for that.

It was an unusual and interesting experience to participate in this lesson, because it was unusual. Even students who hated fractions started to like them.

So Music and Maths are two subjects that became friends.

Edvinas Paškel, 7th form

CHAPTER 4

Mathematics and SportsOrienteering

One of the most active parts of the project was associated with the search for links between sports and math. We organized an orienteering competition. It is a team game in which the team members had to run with each other tied up with ropes and perform a variety of tasks related to mathematics that were hidden in various classrooms. To make it more interesting, tasks were arranges into charades.

Eg. Mathematics classroom charade:

The first-graders need to think a lot,How many is five plus two?School leavers as well try to solve,Piles of equations.Sets, curves, circles ...So what are you waiting for, friends?

What about the third floor?Hurry up, my friend, because others are on their way…

„One beautiful and sunny day, the 11th of May 2012,we organised an orientation competition for Comenius project "Maths is everywhere." Pupils were divided into three teams and had to solve all the mathematical tasks running from one classroom to another.The race was very funny and we learn a lot during them. Students not only tried their strength in mathematics, but also checked if they have good orientation and are strong enough to run. Pupils were solving various tasks and after each correct answer they would get a mark and another task to fulfill. Once students did everything, they ran back to the starting position. All teams were timed. Each participant received a thank you note for participation and sweet gifts were given to the winners. I think that those who participated in this competition, will never forget this day because it was full of fun, laughter and joy of winning! Moreover, we learnt some mathematics for sure.“ Eigintė Kuklytė, 7th form

Film Making

Using footage from various school sporting events, students created a video-task, which not only showed the link between sport and mathematics, but also became a good material for further integrated mathematics lessons.

Posters „Math in Sports“

When looking for links between math and sports Comenius project participants prepared posters that reflected all the information collected. Pupils selected range of sports: cycling, motor sports, athletics, water sports, tennis, winter sports, golf and basketball. The participants were given a task to show mathematics in sports. This work has included the entire school community, encouraging student initiative and creativity. Classes worked individually and searched for materials related to mathematics and sport in a variety of sources. The posters introduced the most important and the most interesting information and facts revealing the links between mathematics and sports. We learned that:

- Football gate aspect ratio must necessarily be 1:3.

- An international competition, where the biggest sum of scored points was reached, was during Asian Games in India in October 1982, when Iraq and Yemen played - 251:33.

- To loose weight a person should exercise at a pulse rate between 0.6 * (220-age) and 0.7 * (220-age) and so on.

Posters were hung in the school lobby, so everyone could read and deepen their knowledge of mathematics and physical interfaces. Later, these posters were presented at a dedicated event.

Maths and Physical Education

Since ancient times people have loved maths and sport. The development of Olympics Games caused the improvement of abstract thinking. Maths was used to show sports results, records and achievements. Today Maths is present in every sport discipline. We would like to show connections between Maths and sport.

“Maths is measure of everything” Aristoteles

Table tennis, also known as ping-pong, is a sport in which two or four players hit a lightweight ball back and forth using. The game takes place on a hard table divided by a net. Except for the initial serve, players must allow a ball played toward them only one bounce on their side of the table and must return it so that it bounces on the opposite side.

Players are equipped with a laminated wooden racket covered with rubber on one or two sides depending on the grip of the player. The table is 2.74 m (9 ft) long, 1.52 m (5 ft) wide, and 76 cm (30 inch) high. The table or playing surface is divided into two halves by a 15.25 cm (6 inch) high net handball. It must be painted green, blue or black.

Handball

The sport is played on a grass field between 90 and 110 meters long, 55 to 65 meters wide. The field has two parallel lines 35 meters from the goal line, which divides the field into 3 sections; each section can have up to 6 players of each team. The goal area is a semicircular line with a 13-meter radius, and the penalty mark at 14 meters from the goal. The goal is 7.32 meters wide and 2.44 meters high.

The game is played with the same ball as the indoor type by two teams of 11 players (plus 2 reserves) and two periods of 30 minutes each.

Indoor handball gradually grew in popularity to replace field handball and the last field handball World Championship was played in 1966.

Each goal has a rectangular clearance area of three metres in width and two metres in height. The goals are surrounded by the crease. This area is delineated by two quarter circles with a radius of six metres around the far corners of each goal post and a connecting line parallel to the goal line. Only the defending goalkeeper is allowed inside this zone.

However, the court players may catch and touch the ball in the air within it as long as the player starts his jump outside the zone and releases the ball before he lands (landing inside the perimeter is allowed in this case as long as the ball has been released). A standard match for all teams of 16 and older has two periods of 30 minutes with a 15-minute half-time.The ball is spherical and must be made either of leather or a synthetic material. It is not allowed to have a shiny or slippery surface.

Volleyball

The ball must be spherical, made of leather or synthetic leather, have a circumference of 65–67 cm, a weight of 260–280 g and an inside pressure of 0.30–0.325 kg/cm2.

The court: A volleyball court is 18 m (59 ft) long and 9 m (29.5 ft) wide, divided into 9 m × 9 m halves by a one-meter (40-inch) wide net. The top of the net is 2.43 m (7 ft 11 5/8 in) above the centre of the court for men's competition, and 2.24 m (7 ft 4 1/8 in) for women's competition, varied for veterans and junior competitions.

A line 3 m (9.84 ft) from and parallel to the net is considered the "attack line". This "3 meter" (or "10 foot") line divides the court into "back row" and "front row" areas (also back court and front court). These are in turn divided into 3 areas each: these are numbered as follows, starting from area "1", which is the position of the serving player:After a team gains the serve (also known as siding out), its members must rotate in a clockwise direction, with the player previously in area "2" moving to area "1" and so on, with the player from area "1" moving to area "6".

Football

The length of the pitch for international adult matches is in the range of 100–110 m (110–120 yd) and the width is in the range of 64–75 m (70–80 yd). Fields for non-international matches may be 90–120 m (100–130 yd) length and 45–90 m (50–100 yd) in width, provided that the pitch does not become square.In front of each goal is an area known as the penalty area. This area is marked by the goal line, two lines starting on the goal line 16.5 m (18 yd) from the goalposts and extending 16.5 m (18 yd) into the pitch perpendicular to the goal line, and a line joining them. This area has a number of functions, the most prominent being to mark where the goalkeeper may handle the ball and where a penalty foul by a member of the defending team becomes punishable by a penalty kick.

A standard adult football match consists of two periods of 45 minutes each, known as halves. Each half runs continuously, meaning that the clock is not stopped when the ball is out of play. There is usually a 15-minute half-time break between halves.

Most modern footballs are stitched from 32 panels of waterproofed leather or plastic: 12 regular pentagons and 20 regular heksagons The 32-panel configuration is the spherical polyhedron corresponding to the trucated icosahedrom; it is spherical because the faces bulge from the pressure of the air inside.

Basketball

Maths is very important in basketball. Basketball is a team sport, the objective being to shoot a ball through a horizontally positioned basket to score points, while following a set of rules.

Usually, two teams of five players play on a marked rectangular court with a basket at each width end.A regulation basketball hoop consists of a rim 18 inches (46 cm) in diameter and 10 feet (3.0 m) high mounted to a backboard. A team can score a field goal by shooting the ball through the basket during regular play.

Javelin Throw

The size, shape, minimum weight,and centre of gravity of the javelin are all defined by IAAF rules. In international competition, men throw a javelin between 2.6 and 2.7 m

(8 ft 6 in and 8 ft 10 in) in length and 800 g (28 oz) in weight, and women throw a javelin between 2.2 and 2.3 m (7 ft 3 in and 7 ft 7 in) in length and 600 g.

Do you imagine Modern Olympic Games without discussing the results and predictions of records? It possible thanks to Maths. But we should not forget that the most important thing in sport is honest competition and people, who devote their lives to sport.

Our student compare Maths to sport, because of the fact that very important in learning maths is regular practise and persistence.

MATHEMATICS AND FOOTBALL

Fotball spectators often clap players who score form the corner or from another position a goal which the opponent’s placement can’t stop.

In this cases the ball moves on a trajectory which differs from the usual one by surprinsingly twisting.

How is this phenomenon explained? To be able to answer this question, we start from the proven by experience truth that a body which falls or it’s freely thrown, if during its moving it’s also spinning, then it won’t move on an ordinary trajectory but on a “false” one.

So, if the player wants to score on the “false” trajectory, he has to hit the ball in order that it would execute a rotating movement.

For the ball to rotate it mustn’t be hit centrally, but to the left or the right of its center, depending on the direction we want it to spin.

The odd trajectory of the spinning body was firstly explained in 1852,by the German physician Heinrich Gustav Magnus(1802-1870) and that’s why this phenomena was called “The Magnus Effect”

This way, the ball that spins is in contact with a certain quantity of air, which spinning produces a circulated current of air.

In the same time, the rotating ball executes a rectilinear motion and this way the air will move on the contrary.

The simple arrows represent the way of the movement of the air spinned by the ball and the double arrows represent the air movement in relative rest.

In the part noted with A(the right part of the ball),the ways of the movements of the two layers of air are opposite, so they get thinner and the resultant speed of the air will get smaller.

In the part noted with B(the left part of the ball),the ways of the movements of the two layers of air are the same, so they meet and the resultant speed of the air will get bigger.

Where the speed of the air is faster there appears a decrease of static pressure which results an absorption effect(according to a law introduced by the Swiss physician Daniel Bernaulli-1700-1782).As a result to the difference of pressure which forms

between the right and the left side(between A and B),the ball moves sideways from the higher pressure to the lower one.

A similar phenomenon appears at the tennis game or table tennis game, when the partners play a game with “cut” or “twisted” balls. It is specified that bowling players know the phenomenon. Coming back to the football game, we must notice that scoring a goal form the corner involves knowing a few laws of physics too.

CHAPTER 5

Maths and The Fine ArtsThe relationship between mathematics and the fine arts is very tight. Mathematical

ideas and the fine arts have been developed simultaneously. Such concepts as time, space and ratio are the base for the fine arts. It is not surprising that our students were examining this relationship according to Comenius project.

During the discussion about our students’ works we would like to tell a few words about main trends in art. They were very helpful during the project realization.

Cubism was one of the most influential visual art styles of the early twentieth century. It was created by Pablo Picasso (Spanish, 1881–1973) and Georges Braque (French, 1882–1963) in Paris between 1907 and 1914. Louis Vauxcelles, the French art critic, called the geometric forms in the highly abstracted works "cubes." Other influences on early Cubism have been linked to Primitivism and non-Western sources. The stylization and distortion of Picasso's ground-breaking Les Demoiselles d'Avignon (Museum of Modern Art, New York), painted in 1907, came from African art. Picasso had first seen African art when, in May or June 1907, he visited the ethnographic museum in the Palais du Trocadéro in Paris.

The Cubist painters rejected the inherited concept that art should copy nature, or that they should adopt the traditional techniques of perspective, modeling, and foreshortening. They wanted instead to emphasize the two-dimensionality of the canvas. So they reduced and fractured objects into geometric forms, and then realigned these within a shallow, relieflike space. They also used multiple or contrasting vantage points.

It is generally considered that cubism has three phases:

Early cubism - (1906 – 1909)

The Girls of Avignon - (Les Demoiselles d’Avignon) by Pablo Picasso.

Synthetic Cubism (1914 – 1919)

Three Musicians by Pablo

Picasso

Abstract art can be a painting or sculpture that does not depict a person, place or thing in the natural world - even in an extremely distorted or exaggerated way. Therefore, the subject of the work is based on what you see: color, shapes, brushstrokes, size, scale and, in some cases, the process. Abstract art began in 1911 with such works as Picture with a Circle (1911) by the Russian artist Wassily Kandinsky (1866-1944).

Kandinsky believed that colors provoke emotions He also assigned instrument tones to go with each color.

“Mit und Gegen” – by Wassily Kandinsky

“Im blau in blue” – by Wassily Kandinsky

„Black and violet”- by Wassily Kandinsky

The mathematical and artistic club made the art works according the following division:

The first task. Students from the first forms made some very interesting paintings. Their works were patterned on the cubism and an abstract art.

The second task. Students from the second forms created some geometric sculptures using different materials like plastic, metal, cardboard, styrofoam.

The third task. Students from the third forms, inspired by maths, designed some unconventional fashion show.

Some paintings made by the students of the first forms

Our students painted some art works that can be characterized as mature, creative, original and full of emotions. The paintings enchant us with a colour range and ingenuity. Only a small part of the paintings is presented below.

Wiktor Kaczmarczyk – form Ib

Emilia Sikora – form IIIe

Daniel Frąk – form I a

Mateusz Golmento – form I

Sylwia Kraińska – form Id

Anna Niebudek – form II b

Sara Skrzyniarz – form I b

The second forms – modern sculptures

Every class had to create an abstract sculpture using the cubism and abstract art ideas. The students’s task was also to use some mathematical concepts, geometric elements and different materials. All the forms managed to do the task and as a result we can admire the following sculptures.

class IIa has created a sculpture called “Cubes”. It amazes with its colours, simplicity and the precision of the origami art.

”Cubes” – form II a

class IIc wanted to draw our attention to the matter that maths is everywhere even though we do not notice it in our everyday life. “The Snail” is made of a tin, some metal sticks and some scraps of cloth. Our students succeeded to prove that you can find and learn maths even when you admire beauties of nature.

”The Snail” – form IIc

class IIb has prepared a three-dimensional sculpture of a clock that shows two ways of understanding the definition of time. On the one hand, time passage of time is inevitable but on the other hand, time is a relative notion. The sculpture is two mutually crossed discs made of styrofoam. The colours of the clock discs harmonize with each other.

”The Clock” – form II b

Students from class IIe have put together geometric elements with some natural images. The sculpture is called ”A Mathematical Tree”. It is a modern sculpture made of metal, wood and plastic that is painted green. A metal construction of a tree is decorated with some elements of geometry like plastic and wooden triangles, a CD, some cardboard boxes, some cylindrical containers, a metal disc and a lot of others.

”A Mathematical Tree” – form II e

Students from class IId created an abstract sculpture that contains number 31 in

it. The most important here

is its symbolic meaning – time flies and takes a roundabout way. It symbolises the end of something old and the beginning of something new.

”The Tower of Time” –form II d

The third forms – a fashion show

Our oldest students had to prepare a fashion show. The main idea was to demonstrate their creativity as well as to awake their imagination. The students managed to design and show their collection that has put together some mathematical ideas, different materials and some fashion trends.

Checks present the set of the similar rectangles. When we add some trousers with a geometric pattern, we can obtain a very interesting result.

Spots that were very fashionable not far ago. They are the perfect geometric shapes – circles. If you join them with the diamond patterned tights, it appears to be a proposal for courageous people.

And more spots again …….. with some striped stockings.

Here is a wide selection of mathematical clothes.

The perfect outfit for the school

disco.

Don’t know what to choose? You can try pink spotted dress or maybe striped trousers and magic glasses. You choose!

CHAPTER 6

MATHEMATICS AND PHOTOGRAPHY

"Photography is a concept,not an accident"

Like it or not,maths has its connections with the compositional rules in photography.Each one of us had heard about the golden number,the golden section,the rule of the thirds,lines and power spots.

The rules are generally applied in art,not only in photography.We can notice some basic compositional rules used in cinematography,painting,architecture,even in our bodies' building,in its proportions and in the proportions of nature,animal and insects too.

There were artists who ignored continuously the rules,like the great American photographer Ansel Adams but you can't say the same thing about Leonardo Da Vinci who spent his whole life being obsessed of these proportions which he considered them to be 'divine' having as an example the famous Mona Lisa and the well known Vitruvian Man.

Leonardo Fibonacci,considered to be a matematics genius of the Middle Age,had determined a sequence of numbers,known as The Fibonacci Sequence and created in 1202.

This sequel of numbers represents the expression of everything which represents proportion in nature simplified to 1,616033 kown as the golden number.

Though this proportion perfect as value,of 1,618 was known in ancient times as the golden section,Fibonacci is the one who determined this number by his sequel.Starting from 0,every number is equal to the sum of the two numbers in front of it and the sequel does not end with 55.This way the golden sections of nature are obtained,strarting with the petals of a flower,the animal body structure and a snails shell constitution.

It was demonstrated that by considering this golden number in art,as a sections or as a proportion,the psychologic impact gets bigger.In photography,cinematography and painting this impact is described by the first impression we have on a certain work of art.Of course that in the process of creating this kind of impact the authot doesn't stop by using only these techniques because,like we said,the rules can be ignored sometimes.

Another important rule is "the rule of the thirds" a well-known concept among artists.

We have a frame splitted in 9 parts and 9 sectors,defined by two vertical and two horizontal lines placed at equal distances.These lines are also named 'power lines",and their intersections are called 'power spots‘All these are imaginary lines and dots,they don't really appear on the photo!

From my point of view,using certain compositional elements help a photography to be more appreciated even though the subject isn't centered in the frame or the skyline isn't placed in the centre of the frame.

Those are frequent mistakes made by beginners!

We have to keep something in mind : what makes a photography beautiful isn't the rules you've respected or not.

Maths and DanceJive

Jive is a dance style in 4/4 time that originated in the United States from African-Americans in the early 1930s. It was originally presented to the public as 'Jive' in 1934 by Cab Calloway. It is a lively and uninhibited variation of the Jitterbug, a form of Swing dance. Glenn Miller introduced his own jive dance in 1938 with the song "Doin' the Jive" which never caught on.Jive is one of the five International Latin dances. In competition it is danced at a speed of 176 beats per minute, although in some cases this is reduced to between 128 and 160 beats per minute.Many of its basic patterns are similar to these of the East Coast Swing with the major difference of highly syncopated rhythm of the Triple Steps (Chasses), which use straight eighths in ECS and hard swing in Jive. the players of swing music in the 1930s and 1940s "Jive" was an expression denoting glib or foolish talk. Or derived from the earlier generics for giouba of the African dance Juba dance verbal tradition.

History of Jive:

Maths in Jive: Circles, curbes, symmetry, rythm, and, the most important , speed. Maths is in every single step.

The Jive is danced to music written in 4/4 time. The same time value (6 beats in 4/4 time) is used for single, double or triple jive. The difference in rhythm is determined by the number of steps, according to the speed of music.Rhythm - Quick, quick, quick, quick, quick, quick.

Rythm, steps:

Dancing lines:

Circle lines of symmetry Their application in Jive

http://www.northamptonshireguide.co.uk/theatres.html

Mathematics and a theatre. It seems that those two terms are completely different and mutually exclusive. But when we think thoroughly about this matter we can notice some common features and connections. We can find mathematics in our everyday life, in each place and situation and even in a theatre.

USING MATHS IN A THEATRE CONSTRUCTION

People started to build the first stone theatre in the fifth century B.C. It was finished in the fourth century B.C. and had to have enough room for 17 thousand people. It was a model for later Greek theatres and it had all the characteristic elements. The round orchestra (diameter 24 metres) was surrounded by the amphitheatre. It was called a theatron. A theatron came to refer specifically to semi-circular, tiered, stone seats for viewing performances. The size of the theatron was 100 metres wide and 90 metres high. It also had 78 rows of seats.

In a modern history some different types of stages were developed:

Arena theatre:A central stage surrounded by audience on all sides. The stage area is often raised to improve sightlines.

Thrust theatre: A stage surrounded by audience on three sides. The Fourth side serves as the background. In a typical modern arrangement: the stage is often a square or rectangular playing area, usually raised, surrounded by raked seating.

End Stage:A Thrust stage extended wall to wall, like a thrust stage with audience on just one side, the front.

A revolving stage: a mechanically controlled platform within a theatre that can be rotated in order to speed up the changing of a scene within a show.

MATHEMATICS AND A THEATRE BUILDINGIn a theatre we can distinguish three types of seats: stalls, circles, boxes. The seats and

rows in a theatre are numbered. What is more to construct a theatre some mathematical calculations are essential.

A theatre seating plan

ELEMENTS OF A THEATRICAL PERFORMANCE

Stage props and sets that are used during the play usually have a shape of a rectangle, a triangle and a polygon.

Costumes, that enrich a performance, are very important as well. They are made by theatre designers. The designers have to measure the actors with the tape-measure in order to fit their costumes.

It is possible to stage a play without music but it is rather difficult to imagine a performance without lighting. The brightness of the spotlights is measured by another unit of measure – lux (lx)

USING MATHEMATICS IN A THEATRICAL PERFORMANCE

o a play is divided into acts and sceneso a drama is divided into five parts: exposition, rising action, climax, falling

action, and dénouement, resolution or catastrophe.

oooooooo

o the number of people who works to stage a play (actors, scriptwriters, producers)

o ticket prices, promotions, discounts,o credit or debit balance of a play, statistics,o posters, placards and tickets are usually rectangular,o the duration of a play (time of the beginning or the end of a play, time of

intervals)

Exposition

Rising action

Climax

Falling action

Dénouement, resolution or catastrophe

o a play name for example: “The Maths of Love” directed by Alicja Albrecht, premiere 02.10.2008 in Warsaw (Studio Buffo)

The example of a poster

This project has been funded with support from the European Commission. This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.