30. Listen again and complete the descriptions.

1. A Latin square is a square in which every number, or symbol, would appear once and only once in each row and column.

2. A Graeco-Latin square is a square in which two Latin squares are superimposed on each other, and such that each cell in the grid is unique.

3. Sudoku is a puzzle to complete a partially completed 9x9 Latin square that contains the digits one to nine in each column and row.

4. The "15 Puzzle is the original sliding block puzzle.

SPEAKING

31. Explain the difference between

• rhombus and parallelogram?

• rhombus and kite?

• acute triangle and obtuse triangle?

• equilateral triangle and isosceles triangle?

• quadrilateral and rectangle?

• an angle and the vertex?

32. Discuss the following questions in small groups or pairs.

1. Which shapes in surrounding things (cars, phones etc.) do you like more: circular and curvy or angular? What’s your favourite shape? Why?

2. Do you like modern art? Can you describe any modern works of art that you either really liked or strongly disliked when you saw them?

3. Do you like round doors and windows like in hobbits’ dwellings? Would you like to live in a house like those in the photos in ex.38.

33. Prepare a short presentation for your classmates on one of the topics below. You may use information from this unit, Further Reading section or do some research on the Internet.

    Euler’s Latin and Graeco-Latin squares Magic figures History and types of magic squares

    Geometry in art and nature Geomagic squares Numerology Geometry in nature

34. Describe the following buildings in as many details as possible. Think about their shape, size, material.

1. Building on the Elbe in Hamburg-Altona, Germany

2. Habitat 67, Montreal, Canada

3. Crooked house, Sopot, Poland

35. Do you agree with the following quotations? Why (not)?

1. "We especially need imagination in science. It is not all mathematics, nor all logic, but it is somewhat beauty and poetry." Maria Montessori

2. "The most painful thing about mathematics is how far away you are from being able to use it after you have learned it." James Newman

3. "Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." Felix Klein

WRITING

36. Your friend came across an interesting article about geometric shapes in art and history. Help him to translate some of the sentences he had difficulty with.

1. Three Musicians is a large painting measuring more than 2 meters wide and high. It is painted in the style of Synthetic Cubism and gives the appearance of cut paper.

2. The Luoshu pattern relates to the ancient Chinese ideal of a perfect world made up of nine divisions, with eight cardinal directions and the Son of Heaven at the centre. Energy of life, Qi, is believed to flow smoothly in such an arrangement of perfect balance.

3. Analytic cubists "analyzed" natural forms and reduced the forms into basic geometric parts on the two-dimensional picture plane.

4. In its material sense, the cube is a symbol of stability. As for its mystic meaning, the cube is symbol of wisdom, truth, moral perfection.

5. Three Musicians emphasizes lively colors, angular shapes, and flat patterns.

6. Instead of an emphasis on color, Analytic cubists focused on forms like the cylinder, sphere and the cone to represent the natural world.