INTERESTING MATH GAMES , APPS and ONLINE SITES

Math in architecture

 Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.

Islamic buildings are often decorated with geometric patterns which typically make use of several mathematical tessellations, formed of ceramic tiles (girih, zellige) that may themselves be plain or decorated with stripes.Symmetries such as stars with six, eight, or multiples of eight points are used in Islamic patterns. Some of these are based on the 'Khatem Sulemani' or Solomon's seal motif, which is an eight-pointed star made of two squares, one rotated 45 degrees from the other on the same centre.

                                                       

An Interesting Story about Snowflakes

    Apart from the fact that no two snowflakes are alike, there is a great symmetry in all snowflakes. The scientific world has not been able to understand scientifically that snowflakes have such a symmetry. When mathematicians examined the snowflakes under the microscope, all snowflakes under the microscope differed. The first studies about snowflakes date back to 1611's. Johannes Kepler was talking about a six-symmetrical shape of snow crystals in one article. 20 years later, Rene Descartes made a more detailed study and mentioned that hexagons also have different shapes.

    American Wilson Bentley, the first scientist to work on snowflakes in a microscopic environment, illustrated the following 800 different models as a result of the examination of the whole sample of snowflakes. Although Wilson Bentley is referred to as an American photographer, in three books he wrote - all of which depict the unique state of snowflakes - he mentions that snowflakes have a symmetrical structure. But a situation like this occurred. Wilson Bentley devoted his life to this question.

 “Can snowflakes have any other match? Does a snowflake have no other equivalent in the universe? "

    Bentley's life, unfortunately, was not enough to understand this. He studied many snowflakes, but they all had a different shape. But in 2006, scientists examined snowflakes falling with special microscopes in the city of Norwich, in the study, which was funded by the State and had a budget of 20 million pounds (46 million TL). Scientists, who started their studies in 2006, finally managed to detect a snowflake on December 3 that was exactly the same as the "Bentley snowflake". Just as follows….

    His words showed how he was affected by the snowflakes.

“I discovered under the microscope that snowflakes are miraculously beautiful. It is a great loss that this beauty is not seen by others and not given the necessary importance. Every crystal is a wonder of design and no design is ever repeated.

Apart from snowflakes, Bentley also photographed all forms of water such as clouds and fog. Called the snowflake man, Bentley was the first American to record raindrop sizes and also one of the first cloud physicists.

    The fractal geometry structure found in snowflakes is one of the most important evidences that show us the mathematical structure of snowflakes. Snowflakes are designed to maintain the Chaos theory according to the "Koch Order". These intertwined snowflakes that stretch forever. 

    The way snowflakes are formed is a branch of interest in chaos theory. Mathematically, each snowflake is formed by a fractal pattern of hexagonal shapes defined as n and r nested. Here n is the number of hexagons and r is its width. American Physics Professor Kenneth Libbrecht is another scientist doing research on this magnificent structure of snow crystals. Working at the California Institute of Technology, Libbrecht took real photographs of snowflakes, revealing the flawless beauty of God's art of creation.

 Exponential coronavirus growth

In these days of emergency for Covid-19 we are bombarded with conflicting news: alarmist messages on the one hand, hymns to normality on the other.

Coronaviruses are a large family of respiratory viruses that can cause mild to moderate illnesses, from the common cold to respiratory syndromes. The virus responsible for the current pandemic is a new strain of coronavirus never previously identified in humans. The contagiousness of this virus can be described through a statistical factor called R0.

The R0 value, or the "basic reproduction number", indicates the average number of infections directly caused by each infected individual since the beginning of the epidemic.  If R0 is 2 it means that on average each infected person will infect two people, so the higher the R0 value, the higher the risk of spreading the virus. If R0 is less than 1 (ie each infected infects less than one person), it means that the infection will tend to extinguish naturally because the number of infected will progressively decrease.

Initially, the spread of the virus could be described by an exponential function.

Exponential growth occurs when the growth rate of a mathematical function is proportional to the present value of the function. It appears initially characterized by a rather slow trend which then undergoes a sudden acceleration.

However, the type of growth previously illustrated do not accurately describe the trend of the situation as they are far from a realistic model.  A logistic function describes a curve whose growth is initially almost exponential, then slows down, becoming logarithmic, to reach an asymptotic position where there is no more growth.  Through this type of function it is in fact possible to observe how, in the case of an epidemic, the presence of healed and deaths decreases the R0, contributing to the slowdown, stabilization and resizing of the phenomenon analyzed.

In concrete terms, it is the behaviors that determine the changes. If lockdowns and restrictive measures had not been implemented, the peak of infections and the number of deaths would have been higher.

The graphs have helped to raise awareness of the extent of the phenomenon and also highlight the progress made by the company over the past few months.

 

When the Earth was measured with a stick

         There was a time when our planet seemed huge. Its true size was first revealed by a simple but ingenious way by a man who lived in Egypt in the third century BC. His name was Eratosthenes and he was an astronomer, historian, geographer, philosopher, poet, theater critic and mathematician.

         One day, while reading a papyrus in the library, he came across a curious note. "Far to the south, at the last borders of Siena, a remarkable thing could be seen on the longest day of the year. On June 21, the shadows of the columns of the temples or a vertical stick diminish as noon approaches. At noon the sun's rays slide to the depths of a well, where, on other days, there is shade. And then, exactly at noon, the columns no longer have shade, and the Sun shines directly in the water of the well. "

         Eratosthenes wondered how it is possible that at the same time a stick from Siena should not have a shadow and a stick from Alexandria, located 800 km north, should show a very clear shadow? The only answer was that the Earth's surface is curved. Not only that, the greater the curvature, the greater the difference in length between the shadows. The sun is so far away that its rays are practically parallel when they touch the Earth.

         The sticks at different angles to the Sun will have different lengths of shadows. For the difference observed between the lengths of the shadows, the distance between Alexandria and Siena should be 7 degrees at the Earth's surface. If you could imagine these sticks stretching towards the center of the Earth, they would intersect at an angle of 7 degrees. Well, 7 degrees means about 50th of the entire circumference of the Earth, 360 degrees.

         Eratosthenes knew that the distance between Alexandria and Siena was 800 kilometers. How did he find out? He hired a man to walk and measure the distance, being able to perform the calculation we are talking about. So, 800 kilometers multiplied by 50, results in 40,000 kilometers. This was to be the circumference of the Earth. Eratosthenes was able to measure the circumference of the Earth using only sticks, eyes, feet and mind, with high accuracy.

       Today we know that the Earth has a circumference of 40,075.017 km at the Equator and a southern circumference of 40,007.86 km.

      If you want to redo the Eratosthenes Experiment, you can do it with 105 other countries, 5877 schools and 36,000 students, by participating in the Eratosthenes Experiment contest https://eratosthenes.ea.gr/.

 Presentasion of experiment

https://youtu.be/Mw30CgaXiQw?list=PL3KYzGGAjjbQoR0YLsCWJyhY-I2EYDH84

                                                                                Consatntina R./ Florinela B./ LTDM Bacau/ Romania 

An interesting story about sum of the first hundred natural numbers

 

Carl Friedrich Gauss was the German  Mathematician is known for his significant contribution in many fields of pure mathematics, like number theory, geometry, probability theory, geodesy, planetary astronomy, the theory of functions, and potential theory.

 

At school we use his form to calculate the sum of the first hundred numbers.

 Tasks of this kind were solved in a geometric way, back in ancient Greece.  Calculating the sum of the first hundred numbers using geometry.

We will show number 1 with a square, number 2 with two squares, number 3 with three squares next to each other and so on.

 

Addition 1 + 2 + 3 + ... + n  will be the arrangement of squares that in turn represent individual natural numbers in the form of an isosceles right triangle with a serrated hypotenuse.

 

In some n-th triangle in a row, we will have a total of 1 + 2 + 3 + ... + n squares and the task was reduced to deciphering that number, that is, the sum. Note that the triangle on each of the legs has n squares. Let us take two such triangles, which are congruent, and connect them along the hypotenuses:

 

We get a rectangle consisting of n (n + 1) squares, which is twice the sum of 1 + 2 + 3 + ... + n.

And so we came to the results:

 

 

Nikola PTS/Marina Nikolic/The First Technical School/Serbia