Meanings


	About Turn This design represents one of the hundreds of ways in which a number of identical squares could be put together. In this particular design six squares make a repeating block.			Amazement Based on the Chevening maze, Kent, which was designed by the second Earl Stanhope (1714 - 86). It differs from earlier mazes because it cannot be solved by simply staying to the right (or left) throughout. In this maze that method merely leads back to the entrance.


	Backgammon The game is not inherently mathematical but is a game of strategy which helps to develop logical thinking, planning and reasoning.			Basketweave The placement of the pieces gives the impression of strips weaving under and over each other. Many complex patterns can be achieved in weaving by passing under and over different numbers of cross pieces.


	Best of Both Whirls Each of the squares is a Baravelle Spiral, which is defined as ‘a straight line design based on connecting consecutive mid-points of sides of regular polygons’. The polygons in this example are squares.			Best of Both Whirls (Cushion) This is a Baravelle Spiral. The triangles appear to form curves as they spiral out from the centre.


	Bunch of Fives Pentominoes are shapes made from 5 identical squares. There are 12 different pentominoes and they can be fitted together in many ways to form a rectangles and other shapes. The afghan also demonstrates reflection by having the tiling reversed as though it is seen through a mirror.			Checkmate Chess is one of the oldest and best-known games in the world. It is not inherently mathematical but is a game of strategy which helps to develop logical thinking, planning and reasoning.


	Chartres Cathedral Maze A circular design on the medieval ‘Chemin de Jerusalem’ pavement maze (more correctly, labyrinth) in the nave of Chartres Cathedral, France, built in 1235. Many believe that walking the path represented a pilgrimage to Jerusalem.			Chromatic Scale Take a long strip of paper and fold it in half, half again and half again. Open it up and crease it firmly on the folds, taking care not to bend them in the wrong direction. Stand the paper on its edge with each fold forming a right angle. Look down on it. What you see is called a Dragon Curve. Unfolding the paper will reveal some folds going in, and some going out. The wall-hanging represents the various stage of folding, with in-folds shown in one colour and out-folds in a second colour.


	Code Comfort A simple substitution cypher. Each letter of the alphabet is represented by a different square. Every letter A is represented by one kind of square, every letter B by a second type, etc. A code like this can be cracked by using the frequency with which letters occur in normal language in combination with intelligent reasoning about known facts of the language.			Counting Pane 100 squares in 10 rows. The first row represents the numbers 1 to 10, the second is 11 to 20, and so on. The colours across the top represent factors. Blue stands for 1, yellow stands for 2, red is 3, green is 4, pink is 5, brown is 6, etc. If a square contains yellow it means the number divides by 2, if it has red then the number divides by 3. If a square divides by 2 and 3 it must divide by 6 so also contains brown. There are many number patterns to be investigated.


	Cubism A cube is a three dimensional shape. It has height, width and depth. A flat surface only has height and width so the shapes here merely suggest cubes. The front face of each ‘cube’ is a square. The top and side are parallelograms which are the same size but slope in opposite directions. Pythagoras’ Theorem is used to work out the relative sizes of the squares and parallelograms.			Cubism (Cushion) A smaller version of Cubism afghan.


	Curve of Pursuit The design has this name because it represents four dogs chasing each other round a field. There are several variations to be found. This one might not be considered completely accurate, by some, as it uses equal angles for all triangles (for ease of construction).			Double Base A representation of binary numbers. We usually count using the decimal system (Base 10). It is a system of counting in tens. It uses a 0 and nine other digits to represent all numbers. There is no specific symbol to represent ‘ten’. We have to use two columns of digits, with a 1 and a 0, to make 10. Binary (Base 2) is a system of counting in twos. It uses 0 and 1; there is no symbol for ‘two’. In a similar way to the decimal system, we have to use two columns of digits, for a 1 and a 0.


	Double Shuffle The lighter stripes cover exactly the same area as the darker stripes. On the other side of the cushion the colours are rearranged so that the centre is light and the outer part is dark, each with the same area.			Double Vision 100 small squares, each being a mixture of two basic colours. Ten colours can be seen on the diagonal and the permutations produce 55 different squares. This can also be used to demonstrate that the tenth triangular number is 55 and that the sum of two consecutive triangular numbers is a square number.


	Double Vision (Cushion) A smaller version of Double Vision afghan, showing all the permutations of 4 colours. It can also show that the fourth triangular number is 10.			Ely Cathedral Maze (Cushion) Based on a maze designed by Sir Gilbert Scott in Ely Cathedral in 1870 - the only pavement maze in a British cathedral. It is unusual because it does not show the typical symmetry of most mazes.


	Equal Parts Fractions from one unit to twelfths. Each of the 12 large squares is divided into a different number of sections. This design has also been published as a poster by Tarquin Publications.			Faultless (Cushion) A fault free shape has no lines cutting right across the shape. This design is in four colours to show that any pattern, map or picture can be coloured so that no touching areas are the same colour, using no more than four colours.


	Fibo-optic This afghan incorporates two different mathematical ideas: a) a representation of a cube b) The Fibonacci Sequence (or Series) in six directions			Fibrenacci Based on the Fibonacci series (or sequence) of numbers. This is named after Leonardo Fibonacci who lived around 800 years ago. He was also known as Leonardo of Pisa. The series is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89........... The afghan design omits the first ‘1’ in the series.


	Fibrenacci (Cushion) A smaller version of Fibrenacci afghan.			Figure of Eight An octagon made up from several other geometric shapes, including isosceles right-angled triangles, trapeziums, pentagons and kites.


	From Square to Eternity Three stages of the development of a pattern. Each piece begins with a small square. The square is surrounded by varying numbers of other squares. It can easily be shown that each new square doubles the area of the previous square and doubling the side of a square gives four times the area.			From Square to Eternity (Cushion) One motif from the afghan. The sizes of the squares are worked out using Pythagoras’ Theorem. The length of the diagonal of one square becomes the length of the side of the next. The diagonal of a square is always approximately 1.4 times the length of the side.


	Give Me a Clew Based on a Roman labyrinth in Caerleon, Wales. Roman mosaics were mostly square. This is a unicursal (one pathway) maze. A 'clew' is a thread such as that given to Theseus, by Ariadne, to guide him out of the Cretan labyrinth,			Granny's Draughtboard Draughts (which is known by other names in different parts of the world) is a traditional game which is not inherently mathematical but such games help to develop logical thinking and planning.


	Granny's Ludo The ability to match spots on a dice with the number of squares on the board is known as ‘one-to-one correlation’. The game is also of value in learning to move counters in a logical and systematic order. It may not be obvious to a small child that it is not unfair for each player to start in a different corner of the board and that it does not make any difference whether there are 2, 3, or 4 players.			Half Measures Q How many different ways can a square be coloured so that half is one colour and half another? A An infinite number. These squares are in blocks of eight and each block shows a different progression or series. In some series the change is made along the diagonal of the square, others are parallel to the sides of the square and some rely on moving whole sections to new positions.


	Have It All Ways 24 squares showing all the possible combinations of four colours. The design demonstrates a systematic way to find all possibilities.			Have It All Ways (Cushion) There are 24 different ways of arranging four colours. 12 of these can be turned upside down to give the other 12.


	Key Rings This is a visualisation problem. It represents a combination lock. The eight rings have to be rotated until a cross is formed in a single colour. There are 128 solutions.			The Long and Winding Road A spiral is ‘any plane curve formed by a point winding round a fixed point at an ever-increasing distance from it’. This design does not fit the definition because it is based on a rectangle not a curve. 'Rectangular spiral' is not a mathematically accurate definition. When the colours are removed it can be seen to be one continuous line which starts at the edge, winds its way to the centre, then returns to the edge.


	Making Waves When viewed from the side the columns are towers with six sides although the original design is based on a regular octagon (8-sided shape). A 2-colour octagon can still be seen at the end of each column. Each column has a row of squares running through the centre with parallelograms on each side. These are proportioned to create the illusion of distance.			Mere Bagatelle This design represents the number of different paths from top to bottom. It is based on Pascal’s Triangle. Although this pattern is named after Blaise Pascal (1623 - 1662) it was known in China around 1300 AD.


	Metafourmosis A quadrilateral is a shape with four straight sides. There are six types of quadrilateral which have special properties and special names. All are included in this design. They are square, rectangle, parallelogram, rhombus, trapezium and kite. There can be many variations of each type.			Meta4 (Cushion) The cushion version of Metafourmosis also includes examples of all quadrilaterals.


	Noughts and Crosses Noughts and crosses (which is known by other names in many countries) is a traditional game, for two people, which is not inherently mathematical but such games help to develop logical thinking and planning strategy Unlike most strategy games, each game is over very quickly and requires much less concentration			Penrose Penrose is a representation of part of a well-known tiling pattern. There are several patterns of this type, known as Penrose Tilings after Sir Roger Penrose who has done a great deal of work on them. These are known as aperiodic, or non-periodic, patterns. At first glance some may appear to be repeating patterns but on closer inspection this is not the case.


	Penrose (Cushion) This is the same as the central section of Penrose afghan.			Pieces of Eight Octagons will not fit together to cover a surface. They can be placed with points touching or sides touching but there will always be gaps. In Pieces of Eight the gaps are four-pointed stars which have 8 equal sides to match the sides of the octagon. Each star is made up from four trapeziums and a square but could be made in other ways.


	Point to Point This design is based on hinged tessellations. Theoretically, the squares could be moved apart to form a pattern of rhombuses (seen as spaces between the squares) and squares. As the squares move around the rhombuses get ‘fatter’ or ‘thinner’. In practice the afghan can only be shown in its fully closed or fully open positions.			The Pythagoras Tree A fractal ‘tree'. Pythagoras’ Theorem states that the sum of the areas of the squares on the two shorter sides of a right-angled triangle is equal to the area of the square on the hypotenuse of the triangle. The black shapes in the design are right-angled triangles. They are also isosceles triangles because they have two equal sides. The two smaller squares are therefore the same size. The design repeats itself with the small squares becoming the larger squares for the next part of the tree, similar to the way a fractal repeats with smaller and smaller copies of itself. It is not a true fractal. Some parts of the fractal have been missed off to make it more closely resemble a tree.


	Pythagorean Ripples 65 squares and 108 triangles, each in 4 different sizes. The design was inspired by an article by Richard Goodman in the March 2000 edition of Mathematics Teaching. It is very similar to Pythagoras Tree, working out in all directions.			Quadrille (Cushion) The pattern is generated from the 8 times table. The answers in the table are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104 .... Add the digits of each answer and keep adding until you get single digits: 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5 ... These numbers show the number of squares you must move across the grid, turning right before each number.


	Revolution Unlike most of our afghan designs, this is not based on sound mathematical foundations. The shape is created by a certain amount of distortion which happens when unequal edges are joined.			Roundabouts A square made from four triangles. So why do we say we follow the path round and round?


	Scaled Up Fold a strip of paper as described for Chromatic Scale. When you can make no more folds, calculate the ins and outs for larger dragons. Dragon Curves are space-filling curves which fit together in many different ways. This design uses dragons of 9 different sizes to cover the surface.			Snakes and Ladders Many skills can be developed with this game: one to one correlation, of spots on the die and squares on the board; addition, to find the new position; subtraction to discover how many places have been lost or gained on the snakes and ladders; advanced mathematicians could even work out the odds against landing on the next snake or winning the game.


	Some Square Over The Rainbow This design demonstrates that the sum of odd numbers gives a square number. There are 7 rainbow colours. The sum of the first 7 odd numbers is 49 (1 + 3 + 5 + 7 + 9 + 11 + 13), which is the square of 7.			Spacecraft This is a representation of a Hilbert Open Peano Curve. It is a space-filling curve. Theoretically the curve covers every point on the surface until nothing is left. This can be imagined in mathematical theory but does not work in real life.


	Spacecraft (Cushion) A smaller version of the Spacecraft afghan.			Square Deal 21 different size squares making up a large square. This is thought to be the smallest number of different squares which can make a larger square. This arrangement was discovered in 1978 by A.J.W.Duijvestijn. Since 1939 it had been thought that 55 was the smallest number.


	Square Deal (Cushion) A smaller version of Square Deal afghan.			Square Is It? How many squares are there here?


	Square Root A small square in the centre is surrounded by another square, with the corners of the first touching the sides of the second. These are then surrounded by a third square and so on until there are seven squares in total. The relative sizes of the squares are worked out using Pythagoras’ Theorem. The length of the diagonal of one square becomes the length of side of the next. Each new square doubles the area of the afghan.			Square Snowflake A representation of a Sierpinski Closed Peano Curve. Giuseppe Peano was the first to describe this type of space-filling curve. Waclaw Sierpinski used his ideas to generate a curve with no beginning and no end. Theoretically these curves cover every point on the surface until nothing is left. This can be imagined in mathematical theory but does not work in real life.


	Stereotype If you look ‘through’ this design you will see that some of the squares appear to be further away than others. Due to the careful spacing the eye perceives some squares to be the background and others to be the foreground.			Striptease A representation of the well-known Cafe Wall Illusion.


	Surface Tension Shapes with equal perimeters are joined to create an undulating surface. The inspiration for this design came from the work of an architect named Tony Wills. Some of his designs are known as D-Forms and consist of various geometric shapes joined together so that they distort each other.			Swirl Without End This is a large Baravelle Spiral, which is defined as ‘a straight line design based on connecting consecutive mid-points of sides of regular polygons’. The polygons in this example are squares. The spiral appears to curve round but there are no curves. It is made entirely from straight lines which make up triangles and squares. The final triangle is truncated to create the overall octagon shape but the design could continue to infinity.


	Take Five 120 squares showing all the possible permutations of five colours. The design shows a systematic way of arranging the colours to ensure they are all included.			Tangram A traditional Chinese puzzle. The pieces of a tangram are usually cut apart and used to construct a multitude of patterns and figures.


	The Other Two Sides The two sides of the cushion show the triangles moved around to prove that ‘the square on the hypotenuse is equal to the sum of the squares on the other two sides’. The same triangles appear on both sides of the cushion. The two squares which are built on the sides of a right-angled triangle are changed for a large square on the back. The area is the same because back and front are the same size.			Tilting at Windmills What makes a pattern? Ask any number of people whether this is a pattern, or not. Some will have no doubt that it is, because the navy triangles make a constant repeating design and the other colours are irrelevant; others concentrate on the colours and will disagree. We do not all interpret things in the same way.


	Tilting at Windmills (Cushion) A smaller version of Tilting at Windmills afghan.			Tower Blocks A cube is a three dimensional shape. It has height, width and depth. A flat surface only has height and width so the shapes here merely suggest cubes. Each large cube contains 8 small cubes, though only parts of 7 can be seen.


	Tri-square 36 squares - one in the first colour, two in the second, three in the third, etc. Adding these numbers together gives the sequence of triangular numbers. When you get to add 8 the whole shape forms a 6 x 6 square - square and triangular numbers coincide. If you were to keep going, when would this happen again?			Trivial Pursuit (Cushion) A smaller version of Curve of Pursuit afghan.


	Try Angulate 225 circles in two triangles. 120 form a triangle of plain circles, 105 form a triangle of circles with dark centres. When the two triangles are put together they form a square. Any two consecutive triangular numbers form a square number.			Under Orders The ten colours represent ten cards from a pack. The first column shows their order at the start. When the cards are shuffled in a ‘perfect shuffle’ the cards in the bottom half of the pile alternate with those in the top half. The new sequence is represented in the second column. Each ‘perfect shuffle’ changes the order of the cards and after the tenth shuffle they have returned to their original positions.


	Walls of Troy A 12-sided design based on a maze (more correctly, a labyrinth) in East Yorkshire dating from 1815. The name may have arisen because the city walls of Troy were deliberately confusing. The path of the maze is similar to many others, including that in Chartres Cathedral.			Wind Up An Archimedean Spiral moves out from the centre by the same distance on each rotation.


	Window Boxes There are many different things to be seen in this illusion. Make sense of one part and something else goes wrong. Should the boxes be going away from us in opposite directions? Are we looking at boxes with windows in front , or are we looking at square tubes with a thick purple end section? Can a square be the top of two boxes at the same time?			Windows 100 All the possible combinations of ten colours where one is used for the centre of the square and another for the surround. The arrangement is systematic to ensure that no combinations are missed. Every colour appears inside and outside every other colour. The design can be seen as ten columns with windows which have bars of the same colours passing behind them. The bars are seen through the windows.


	The Woolsack This afghan consists of 13 squares though only 12 can be seen. The 13th is on the back and the rest of the afghan can be folded and stored inside, as a cushion.			Wreck Tangled The 14 green rectangles all have the same area and the total area of the rectangles is the same as the area of the background.


	Wreck Tangled (Cushion) This cushion has 7 rectangles, 3 on one side, 3 on the other, and one that spreads onto both. Each rectangle has the same area and the total of their areas is the same as the area of the background.


	Data Log This afghan demonstrates where triangular numbers and square numbers coincide. Each coloured strip is one unit longer than the strip before. It represents the 49th triangular number and 35th square number. It is closely related to Tri-square cushion.			Click here for information about the construction of afghans and cushions.