WORKSHOP - EVERYTHING IS A NUMBER

We present here Polish students activities within Interactive workshop for students - Interactive Exhibition "Everything is a Number" on Jagiellonian University in Krakow.

TRIANGLES

TRIANGLES

Take a triangle on a handle and place it in a beam of light.

Move the triangle so that its shadow is aligned with the grid lines.

The grid is made up of equilateral triangles of equal size. The triangles on the handle that you can use, have different shapes, but each of them can be set in such a way, that its shadow matches the grid. In projective geometry, all triangles can be converted to each other. You can always choose a perspective from which two triangles will look the same.

 

PROJECTION

PROJECTION

Take one of the polyhedrons on a handle and place it in a beam of light

Try to set it in such a way that the shadow fits to one of the shapes on the grid

 

The grid is composed of different figures. There are triangles, squares, and hexagons. Can some polyhedrons be set so that their shadows fit to the different figures ?

Although the solids used in this experiment are not complicated, their projections may have different shapes, for example, the projection of the cube can be a square, rectangle, or even a hexagon.

MOBIUS BAND

Mobius Band



First instruction:

  • Take a pawn and place it on a metal strip
  • Move it along the strip all the time in one direction and observe when the pawn is returned to the starting point.

The rectangular belt twisted by 180 degrees, and then glued at the ends is called the Mobius band. Its most distinctive feature is that it only has „one side”. Thus the pawn runs on the band before returning to the starting point which was on the „other side”. The band was described independently by both August Mobius (1790-1868) and Johann Benedict Listing(1808-88). The Mobius band has practical applications such as transmission belts or printer tapes.

 

 

 

 

Second instruction:

Fasten the zipper and connect the ends of the belt together to create a loop. Think about how many parts you get when you unzip the zipper.

Fasten the zipper and connect the ends again, this time turning one end of the belt 180 degrees. Think about what happens after unzipping

PYTHAGORAS' THEOREM

A PYTHAGOREAN PUZZLE

Pythagoras' Theorem



Twenty – five identical cubes can be arranged to fill:

either one square with a side length equal to five,

or two squares of side length equal to three and four

In this way, by means of the puzzle, you can prove Pythagoras' Theorem.

 

A Pythagorean Puzzle

 

You can put four identical right-angled triangles into a frame in two ways:

in the first position, the remaining empty field in the frame is a square whose side length is equal to the length of the hypotenuse of the triangle,

in the second position, remaining empty space in the frame comprises two squares built on the other two sides of the triangle.

Such proof of Pythagoras' Theorem was known in antiquity. Historian of science believe that this is the way the theorem was proved by the Pythagoreans.

 

PUZZLE W - M and ENIGMA PUZZLE

W – M PUZZLE

Disconnect the two parts of the puzzle. Do it without the use of force !

ENIGMA PUZZLE

Disconnect the two parts of the puzzle. Do it without the use of force !

Similar puzzles have been known since the 18th century. They were made by blacksmiths for use by their friends as fun, and at the same time were good practice for their apprentices.

How it is this possible ?

Each of the rods has a radius R and a diameter D = 2R, and the distance between the bars is S. Whilst sliding the centres of the bars, form a square with a side lenght equal to D. However, the distance between the centres of the rods belonging to the same part is equal to the lenght of the diagonal of the square. And the minimum distance between the rods which enables them to be untied would  be equal to:

 

Than if the distance between the roods is grater then 0,4D than after 90 degrees twist of the part of the puzzle is could disconnected.

MAGIC SQUARE

Magic Square

Arrange the plates with numbers in such a way that the sum of the numbers in each row, each column and each diagonal is 34.

Magic square were known in antiquity. The magic square, which indicates numbers from 1 to 9 , was described by Loh-Shou aroung 2800 BC. In Europe, this type of Puzzle appeared in the 14th century. This square, which should be arranged here, is one of the most well-known. In 1514, Albrecht Durer (1471-1528) put it on a copperplate entitled Melancholia I.

There is only one magic square with the dimensions 3x3(excluding those obtained by rotation and reflection). In 1693, a list of all 880 4x4 squares was published, and squares with the dimensions 5x5 were counted in 1973, there are 275305224 of them. The number of 6x6 and larger magic squares has not yet been calculated.

MIRROR TELESCOPE

Mirror Telescope

 

Point the telescope at the image placed on the ceiling and try to read the inscription located there.

The miniature telescope used in this exhibit is built in the system proposed by Sir Isaac Newton (1643-1727). Its optical system consists of two mirrors: a parabolic primary mirror and plane secondary mirror, which reflects the light to the side of eyepiece. Newton built his telescope in 1668 and it was the first telescope which used a mirror. Mirror telescopes with mirrors whose shapes are paraboloidal or hyperbolical, are still essential instruments for astronomical observation.

HYPERBOLOID

Hyperboloid

 

Rotate the lower circle and see the shape of the strings.

If you rotate the wheel every string is tilted, but it is still straight. However, looking at the surface created by the strings, you can see a curvature.

The shape of the surface is a hyperboloid of one sheet. This surface can be constructed with straight lines. It is characterised by high resistance to twisting. This is why it is used in architecture – for example, in the construction of power plant cooling towers.

 

PASCAL'S TRIANGLE

Pascal's Triangle

  • Using magnetic pins, cover all the numbers divisible by 2 on a board
  • Notice what geomtric structure was created
  • You can repeat the experiment using numbers divisible by 3 or 5, as wellas other numbers.

 

 

 

 

 

 

The numbers in circles on the board from Pascal's triangle. It is produced in such a way that each number is the sum of two numbers directly above it. At the same time, every n-th row contains the coefficients of the development of Newton's binomial (a+b) to the power of n. The numbers in Pascal's triangle have many interesting properties, such as the sum of the numbers in each row which is equal to the proper power of two. Another interesting feature is the fact that after the removal of even numbers from the triangle, the remaining numbers from the geometric structure of the Sierpiński sieve. A similar reglarity occurs for other natural numbers.

Historians of science believe that the triangle was known in 12th century. Its discoverer is regarded as the Persian mathematician Omar Khayyam (1048-1131). During those times, the triangle was also known in India and China. In the 17th century, French mathematician Blaise Pascal (1623-62) published a treatise in which he described the properties of the triangle and joined with them the results of his research on probability. This is why it is called Pascal's triangle.

How we can solve the problem with the Graph ?

Each dice can be described by a graph in which the nodes represent the four colours of the walls, and the edges conect the colours which are on the opposite sides. If the opposite sides are the same colour we make a loop around the point. The cubes may be represented by four graphs.

The joining of graphs allow a simultaneous representation of colours on the 12 pairs of opposite walls. As the solution requires that the cubes are arranged one on the other, only 8 pairs will contribute to the solution.

To find the solution, extract from this complex two graphs, one corresponding to the walls of the front-to-back, and the other for the right-left walls.

The graphs must takes into account all colours and all four cubes ( or edges numbered from 1 to 4).

In addition, only two lines can come out of the pointsrepresenting the colours. The graphs which fulfil these conditions are shown above.