The Mathematics of Tessellation 2000 Andrew Harris

Transcription

1 The Mathematics of Tessellation 2000 ndrew Harris The Mathematics of Tessellation ndrew Harris

2 ontents Defining Tessellation... 3 Tessellation within the Primary School urriculum... 3 Useful Vocabulary for Tessellation... 3 Prior Knowledge Required for Understanding the Mathematics of Tessellation... 4 Useful Resources for Teaching Tessellation...5 Progression in Learning about Tessellation Tessellation by Repeated Use of One Regular Shape Tessellation by Repeated Use of Two or More Regular Shapes Tessellation of Triangles and Quadrilaterals Preparatory Knowledge for Understanding the Tessellation of Triangles and Quadrilaterals...11 (a) Tessellating with Triangles (b) Tessellating with Quadrilaterals Tessellation of Irregular Shapes obtained by Mutation of Tessellating Shapes...18 The designs of M.. Escher Tessellations involving other irregular shapes Use of IT in Teaching Tessellation Resources Hexagonal Tiling Mats Square Tiling Mats Square Tiling Mats Square Tiling Mats Useful References: Tessellation...29 The Mathematics of Tessellation ndrew Harris

3 Defining Tessellation tessellation can be defined as the covering of a surface with a repeating unit consisting of one or more shapes in such a way that: there are no spaces between, and no overlapping of, the shapes thus employed, and the covering process has the potential to continue indefinitely (for a surface of infinite dimensions). Tessellation within the Primary School urriculum Primary school tessellation activities fall into two categories: The mathematics of tessellation pplication of knowledge about tessellation The application of tessellation aspect is often seen in schools and features in most published mathematics schemes. However, the mathematics of tessellation aspect is often overlooked. This results in poor progression with children often repeating the pattern-colouring activities they have undertaken in previous years of their schooling. Moreover, such children have little idea of why some combinations of shapes tessellate while others do not. The mathematics of tessellation aspect should focus on providing children with the knowledge and skills to explain why tessellation is or is not possible for particular units of shape. It is this which is the basis of good practice for teaching tessellation. That said, it is very worthwhile to show children real-life examples of the application of tessellation. rick walls, paving, wall and floor tilings, woodblock floors, carpets, wallpapers, wrapping papers, textiles and works of art are often useful resources for whole-class or group discussions about tessellation as a concept and about its application in everyday life. Typical activities when using these might be to identify the repeating unit of shape used to create the tessellation, to explain why the tessellation works, to consider other ways of tessellating the given surface or to develop mental visualisation skills (e.g. as part of a mental/oral starter or plenary within the daily mathematics lesson). Using resources of this kind demonstrate to children the value and purpose of understanding the mathematics of tessellation and thus provide an incentive and reason to learn about it. Useful Vocabulary for Tessellation In common with other aspects of Shape and Space work, children will need to become familiar with the following mathematical vocabulary Plane: two-dimensional or, colloquially, flat. Regular shape: a shape in which all the sides are the same length ND all the angles are the same size. Irregular shape: a shape in which not all the sides are the same length ND/OR not all the angles are the same size. Polygon: a two-dimensional, closed shape in which all the sides are straight lines. Interior angles: the angles inside the boundary of a shape (see illustration below). Exterior angles: the angles through which one would turn at the corners of a shape if walking around the boundary of the shape (see illustration below). The sum of the exterior angles of any polygon is 360 or one whole turn. Re-entrant angle: an interior angle of more than 180. ongruent shapes:shapes which are identical in terms of type of shape, lengths of sides and sizes of angles. Their position in space and their orientation may be The Mathematics of Tessellation ndrew Harris

4 different. hildren will also need to know the names and properties of common two-dimensional shapes. Exterior angles Interior angles interior angle + corresponding exterior angle = 180 Interior and Exterior ngles of a Regular Pentagon Prior Knowledge Required for Understanding the Mathematics of Tessellation In order to be able to understand why some combinations of shapes will tessellate and others will not, children will need to know: a whole turn around any point on a surface is 360 ; the sum of the angles of any triangle = 180 the sum of the angles of any quadrilateral = 360 how to calculate or measure the interior angles of polygons The interior angles of regular polygons (i.e. not other polygons) can be calculated in one of two ways: 1 Divide a whole turn (360 ) by the number of exterior angles (= the number of sides) to find the size of one exterior angle. Then use the fact that the exterior angle + the corresponding interior angle = 180 (because angles on a straight line add up to 180 ) to find the interior angle. e.g. for a regular pentagon (5 sides, so has 5 exterior angles) the exterior angle = = 72 so the interior angle = = The sum of the interior angles of a n-sided regular polygon = (n - 2) 180. Once the total has been calculated in this way, the size of one of the interior angles can be found by dividing by the number of interior angles (= n). e.g. for a regular pentagon (5 sides, so has 5 interior angles) The Mathematics of Tessellation ndrew Harris

5 the sum of the interior angles = (5-2) 180 = = 540 so one interior angle = = 108. Note that these methods only work for regular polygons. Measuring the angles with a protractor is possible in all cases, regular or irregular. In summation, children will need to know or learn about the angle properties of all regular polygons and of common irregular polygons in order to understand the mathematics of tessellation. Useful Resources for Teaching Tessellation The following may be found useful when teaching about tessellation: a large number of various cardboard or plastic regular plane (i.e. flat/2d) shapes each of which have sides of the same length; gummed paper shapes which have sides of the same length; tiling mats (obtainable from the ssociation of Teachers of Mathematics or see Resources section at the end of this booklet) computer software packages which allow children to investigate tiling and tessellation activities; real-life examples of tessellation patterns (such as works of art (e.g. by Escher), fabrics, wrapping papers, wallpapers, floor and wall tilings, brickwork patterns); sets of different kinds of triangles sets of different kinds of quadrilaterals sets of irregular shapes (some of which tessellate and some of which do not) protractors Progression in Learning about Tessellation Several distinct stages can be identified in learning about the mathematics of tessellation. s progress is made through these stages the degree of regularity of the shapes under consideration reduces. The suggested stages in the progression are: 1 Tessellations involving repeated use of ONE regular polygon; 2 Tessellations involving repeated use of a unit of shape made up of TWO OR MORE different regular polygons; 3 Tessellations involving triangles or quadrilaterals; 4 Tessellations of irregular shapes obtained by transformation of other more regular, tessellating shapes ; 5 Tessellations involving other irregular shapes. The Mathematics of Tessellation ndrew Harris

6 1. Tessellation by Repeated Use of One Regular Polygon Tessellations in which one regular polygon is used repeatedly are called regular tessellations. Initially, children should investigate using sets of 2D shapes to find out which common regular polygons will tessellate and which will not. They should arrive at the following conclusions: Equilateral triangles will tessellate: Squares will tessellate: Regular hexagons will tessellate: No other regular polygons will tessellate in this way: for example, regular pentagons regular heptagons regular octagons The Mathematics of Tessellation ndrew Harris

7 ll of the above conclusions can be reached by simply trying each regular shape in turn. It is much harder, however, for children to be able to explain why it is that, of all the regular shapes, only the equilateral triangle, the square and the regular hexagon will tessellate. t this point the prior knowledge outlined on page 4 must be utilised. The key piece of knowledge here is that a whole turn around any point on the surface is 360. The vertices of six equilateral triangles meet at Point. Each of the interior angles of the equilateral triangles is Point at which the vertices of the six triangles meet. ngle sum around Point is = 6 60 = 360 The sum (total) of the angles around Point is 6 60 = 360. This fact is true of all such points where the vertices of six equilateral triangles meet and thus the equilateral triangles will tessellate. n alternative way to look at this idea is to note that 6 complete equilateral triangles can meet at a common vertex at any point on the surface to be covered (because = 6, a whole number) and without any gaps being left or any overlapping occurring (because 360 is exactly divisible by 60 ). [Note that it does not matter what size the equilateral triangles are (as long as they are all congruent) since the angles will still be 60 whatever the length of the sides.] This tessellation may be represented by the abbreviated notation 3 6 (signifying that six threesided regular polygons meet at a common vertex). Note this does NOT mean three to the power six in this context. You may wish to avoid this notation when working with children because of the potential for confusion with the more usual interpretation of this as three to the power six. It is included here because of its usefulness for the speedy notation of tessellation patterns involving regular shapes. The same idea can be applied to the tessellation of squares and of regular hexagons: Point at which the vertices of the four squares meet ngle sum around Point is = 4 90 = 360 The total of the angles around Point is 4 90 = 360. Since this is true of all such points where the vertices of four squares meet this explains why squares tessellate. lternatively, the four complete squares can meet at a common vertex at any point on the The Mathematics of Tessellation ndrew Harris

8 surface (because = 4) and without any gaps being left or any overlapping occurring (because 360 is exactly divisible by 90 ). This tessellation may be represented by the notation 4 4 (four four-sided regular polygons meet at a common vertex). Point at which the vertices of the three hexagons meet. ngle sum around Point is = = 360 The total of the angles around Point is = 360. Since this is true of all such points where the vertices of three regular hexagons meet this explains why regular hexagons tessellate. lternatively, three complete regular hexagons can meet at a common vertex at any point on the surface (because = 3) and without any gaps being left or any overlapping occurring (because 360 is exactly divisible by 120 ). This tessellation may be represented by the notation 6 3 (three six-sided regular polygons meet at a common vertex). However, if we use a similar line of argument for the other regular polygons we find that: Regular Shape Interior ngle Size 360 Interior ngle Tessellates? Regular Pentagon = No Regular Heptagon = No Regular Octagon = No Regular Nonagon = No Regular Decagon = 2.5 No etc Note that, for each shape in the table above, the result of dividing 360 by the interior angle is not a whole number. onsequently, for any of these shapes it is impossible for an exact (whole) number of them to meet at any point on the surface to be covered. Thus, either gaps will be left between them or overlapping of the shapes will occur and therefore none of these shapes can be used to create a regular tessellation. Of all the regular polygons, only the equilateral triangle, the square and the regular hexagon have interior angles such that the result of dividing 360 (a whole turn) by the interior angle is a whole number. onsequently, only these three regular polygons can be used to create regular tessellations. The Mathematics of Tessellation ndrew Harris

9 2. Tessellation by Repeated Use of Two or More Regular Polygons Having explored the tessellating possibilities of single regular polygons children should be asked to investigate which combinations of two or more regular polygons will tessellate and then consider why some combinations are successful and others are not. Tessellations in which: there are two or more regular polygons around each common vertex, the tiling around each common vertex is identical are known as semi-regular tessellations. hildren should be taught to identify the repeating unit (composed of 2 or more shapes) in such tessellations. This builds upon the skill of being able to identify the repeating unit in linear patterns such as those made with beads on a string or with multilink which children should have already experienced in previous work. There are 8 semi-regular tessellations to be found. Each is shown below with the abbreviated notation signifying how many of which type of regular polygon are located around each common vertex. and (3 equilateral triangles & 2 squares) (3 equilateral triangles & 2 squares) ngle sum around common vertex ngle sum around common vertex = = 360 = = (4 equilateral triangles & 1 regular (2 equilateral triangles & 2 regular hexagon) hexagons) ngle sum around common vertex ngle sum around common vertex = = 360 = = 360 The Mathematics of Tessellation ndrew Harris

10 4.8 2 (1 square & 2 regular octagons) (1 equilateral triangle & 2 dodecagons) ngle sum around common vertex ngle sum around common vertex = = 360 = = (1 square, 1 regular hexagon, 1 dodecagon) (1 equilateral triangle, 2 squares & 1 regular hexagon ngle sum around common vertex ngle sum around common vertex = = 360 = = 360 For each of these semi-regular tessellations, the sum of the angles around each of the common vertices is 360 and this is the reason why each of these combinations of regular polygons produces a viable tessellation. Other combinations of regular polygons do not produce semiregular tessellations because it is not possible to achieve an angle sum of 360 around each common vertex while maintaining identical tiling around each common vertex. The Mathematics of Tessellation ndrew Harris

11 3. Tessellation of Triangles and Quadrilaterals Up to this point all work undertaken has been with regular polygons. Once the idea has been established that the viability of a potential tessellation is determined by the angle sum around the common vertices of the shapes involved, the child can begin to explore the tessellating propensities of other polygons which are less regular. Two important sets of irregular polygons that should be investigated are triangles and quadrilaterals. Preparatory Knowledge for Understanding the Tessellation of Triangles and Quadrilaterals hildren will need to learn that: the sum of the angles in a triangle is 180 ; the sum of the angles in a quadrilateral is 360. The Sum of the ngles in ny Triangle The fact that sum of the angles in a triangle = 180 is usually proved to children by asking them to draw on paper any old triangle with angles, and. The triangle is cut out with scissors and the corners of the triangle are then torn off and arranged as shown below: tear off corners & arrange on a straight line The fact that the angles, and can be arranged to lie on a straight line (check with a ruler) indicates that for this triangle the sum of the angles is equal to a half-turn or 180. If several children attempt this each with different triangles it can be shown to work for several triangles. This is then usually accepted as adequate evidence that the sum of the angles of any triangle = 180. Note that this procedure does not constitute a rigorous mathematical proof of this mathematical statement since there are an infinite number of possible triangles and therefore not all triangles have been tested by the procedure outlined above. However, it is usually considered an adequate basis upon which to proceed for children at this level of mathematics. The visual nature of the proof helps to convince children of the truth of the hypothesis. The Sum of the ngles in ny Quadrilateral The fact that the sum of the angles in any quadrilateral = 360 can be proved in a similar way to the procedure outlined above for triangles. Draw any quadrilateral, tear off the corners and arrange around a common point: The Mathematics of Tessellation ndrew Harris

12 tear off corners & arrange around a common point D D Then use the fact that a whole turn = 360 to deduce that D = 360 and so prove that the sum of the angles of any quadrilateral = 360. Note that the same reservations expressed above (regarding the procedure for proving the angle-sum of triangles) about lack of mathematical rigor apply in this case also. Triangles and quadrilaterals can be explored in relation to their tessellating propensities by either presenting the task as a pair of investigations (i.e. Which triangles tessellate? Which quadrilaterals tessellate?) or as a pair of hypotheses ( ll triangles tessellate, ll quadrilaterals tessellate ) which children are asked to test and so either prove them or disprove them. (a) Tessellating with Triangles n initial exploration of a few different types of triangles leads one to suppose that most triangles will tessellate: Different types of isosceles triangles The Mathematics of Tessellation ndrew Harris

13 Right-angled triangles Scalene triangles In fact, any triangle can be used as a repeating unit with which to tessellate. In order to prove this, children can make use of what they already know, namely, a whole turn about any point on the surface is 360. In addition to this, children will also need to know that the sum of the angles of any triangle is 180 (see earlier section Preparatory Knowledge for Understanding the Tessellation of Triangles and Quadrilaterals ). If it can be established that, for any triangle, the sum of the angles around any common vertex is always 360 this will prove that all triangles tessellate. This can be done simply with children by asking them to draw with a ruler any triangle with angles labelled, and. This triangle is then replicated and the triangles arranged so that around any common vertex there will be from the various triangles involved two of each of the angles, and. We know that + + = 180 (because the sum of the angles in any triangle is 180 or, alternatively, because, and form a straight line) The Mathematics of Tessellation ndrew Harris

14 and around any common vertex the sum of the angles = ( + + ) + ( + + ) = = 360 Thus, any triangle can be used as a repeating unit for tessellating. (b) Tessellating with Quadrilaterals similar exploration of the tessellating properties of different types of quadrilaterals can be undertaken. s a result of the properties of various quadrilaterals, there may be more than one way in which a particular quadrilateral may tessellate. For example, there are many ways of tessellating with rectangles. These are just a few: rickwork patterns are often a rich source of everyday examples of such tessellations. Initial explorations involving quadrilaterals suggest that, like triangles, most quadrilaterals can be used as a repeating unit with which to tessellate: Tessellating with a rhombus The Mathematics of Tessellation ndrew Harris

15 Some other possible tessellations using a rhombus When tessellating with a parallelogram, children are likely to discover that they can produce similar tessellations to the first two rhombus tessellations given above. trapezium or kite will also tessellate: Tessellating with a trapezium Tessellating with a Kite It is also worth asking children to investigate quadrilaterals which have re-entrant angles such as the dart which also tessellates: tessellating with a dart fter some exploration, children should begin to form the hypothesis that all quadrilaterals can The Mathematics of Tessellation ndrew Harris

16 be used as a repeating unit with which to tessellate. This hypothesis can be proved geometrically in a similar way to that for triangles. gain, it relies on the fact that a whole turn around any point on the surface to be tessellated is 360. This time, however, children will also need to know that the sum of the angles in any quadrilateral = 360 (see earlier section Preparatory Knowledge for Understanding the Tessellation of Triangles and Quadrilaterals ). If it can be established that, for any quadrilateral, the sum of the angles around any common vertex is always 360 this will prove that all quadrilaterals can be used to tessellate. hildren are asked to draw any quadrilateral using a ruler and pencil. D This is then used to tessellate in such a way that around any common vertex there will be (from the 4 quadrilaterals which meet there) one of each of the angles,, and D as shown below: D D D D D D D D D D D D D D D D The Mathematics of Tessellation ndrew Harris

17 Given that we know the angle sum of any quadrilateral to be 360 then D = 360 Since the angles around any common vertex are precisely,, and D and so must total 360, we can then state that all quadrilaterals can be used as a repeating unit with which to tessellate. tessellation using a combination of two quadrilaterials. The repeating unit contains 3 kites and 1 rhombus. The Mathematics of Tessellation ndrew Harris

18 4. Tessellation of Irregular Shapes obtained by Transformation of Other Tessellating Shapes It is possible to produce some very irregular shapes which will tessellate by transforming other shapes which are known to tessellate. These irregular shapes include shapes bounded by curved lines (up until now only shapes with straight sides have been considered). Squares, rectangles, equilateral triangles and hexagons are suitable shapes from which to start. y translating or rotating about the mid-point of any side sections of the starting shape (or a combination of both) a new, irregular shape can be made which will also tessellate. Some examples of this are given below: Translating a Section of a Square The original square section of the original square is moved by translating (sliding) it to the opposite side of the square. The resulting tessellation Rotating Sections of a Square The original square section of the original square is moved by rotating it about the midpoint of the side of the square. similar rotational process is applied to another side of the square. The resulting tessellation The same methods can be applied to rectangles, equilateral triangles and regular hexagons. The Mathematics of Tessellation ndrew Harris

19 To explain why these irregular shapes tessellate is more difficult than for previous cases encountered in Sections 1-3 and is usually considered to be beyond the level expected of primary school children. The following explanation is included for subject knowledge purposes. The argument that, around any common vertex within the tessellation, the sum of the angles is 360 is still true but this is now complicated by the fact that the boundary of the repeating unit may include curved lines. Where curved lines are involved, we use the tangents to the curves to define the limits of the angles: d a c b The dotted lines are the tangents to the curves at the common vertex. The angles a, b, c and d are the angles between the tangents (dotted lines) and as before a + b + c + d = 360 Practical Work with hildren The designs of M.. Escher owe much to this idea of transformation of shapes to create new, irregular, tessellating shapes. Examples of Escher s work are useful resources for inspiring children to create their own designs in a similar style. Escher s Lizard Tessellation: hexagonal grid has been superimposed on this tessellation to show how it has been created. It uses a replicating Lizard tile made from a regular hexagon which has undergone several mathematical transformations. The Mathematics of Tessellation ndrew Harris

20 Left: Escher s Flying Horse tessellation Right: Escher s irds tessellation The simplest way in which to do this type of work with children is to use a gummed paper shape as the initial shape. Sections of this gummed paper shape can then be cut out and either translated or rotated appropriately into their new positions. ll of the pieces of gummed paper can then be stuck to cardboard. utting around the outline of the transformed gummed paper shape will create a template. This can then be used as the repeating unit for a tessellation. Effective displays of children s work can be created if colour is used to emphasise the different arrangements of the repeating unit within the tessellation. The Mathematics of Tessellation ndrew Harris

21 5. Tessellations involving other irregular shapes t this stage, it becomes sensible to widen the range of shapes to include all irregular shapes. Some examples of activities might be investigating irregular polygons: e.g. irregular pentagons or or (This is useful to consider with children in order to avoid/confront the misconception that pentagons don t tessellate which arises when children over-generalise the non-tessellation of regular pentagons and thereby assume that it is impossible to tessellate with any pentagon). There are, currently, 14 known irregular pentagons which tessellate but no-one, to date, is certain if any more exist.... using letters of the alphabet: H-shapes Y-shapes E-shapes The Mathematics of Tessellation ndrew Harris

22 ... or shapes derived from circles: e.g. using to produce... or to investigate the tessellating possibilities offered by the various sets of polyominoes (extension of the idea of the domino): The set of triominoes The set of tetrominoes Shapes composed of Shapes composed of 3 squares 4 squares ll of the triominoes and the tetrominoes can be used as a repeating unit with which to tessellate. Shapes composed of 5 squares. Only some of these can be used as repeating unit with which to tessellate. The set of 12 pentominoes The Mathematics of Tessellation ndrew Harris

23 Hexominoes are composed of 6 squares (there are 35 of these). Each of these will tessellate. Similar tessellation investigations can be carried out with polyiamonds (similar to polyominoes but made of equilateral triangles instead of squares): The set of triamonds (just one!) The set of tetriamonds etc. Use of IT in Teaching Tessellation There are several IT software packages which address aspects of tessellation. Most common are dedicated tiling packages which allow children to select from a range of different polygons and use them to tessellate, thereby using the computer screen as the surface to be covered. The advantage of using such a software package to do this is that it is relatively easy for a child to obtain a paper-based record of his or her work via a printer and it bypasses the tedious and time-consuming aspects of tessellating (drawing round templates and colouring in). There are obvious advantages in relation to presentation of children s work as well. If the computer is linked to a large monitor, a large TV or a data-projector then a software package may be used as a resource for direct teaching and discussion (perhaps within the mental/oral starter or plenary of the daily mathematics lesson). However, there are also some disadvantages involved with using IT packages for this purpose. One is that it is often difficult to measure and/or angles within shapes on a computer screen. the screen size and resolution may also be a problem. Most packages limit the types of shapes available to children (often to just regular polygons and perhaps a few common irregular shapes such as rectangles). In some packages the emphasis is on making patterns (i.e. the application of tessellation to design) rather than on the mathematical aspects of tessellation. oth of these areas of knowledge have their value but it is important to identify which of these aspects a software package addresses. The best tiling packages allow the user to design their own tile or repeating unit of shape(s). The very best of these allow the user to translate or rotate portions of tessellating shapes so as to form new, irregular, tessellating shapes (by the procedures outlined previously in the section Tessellation of Irregular Shapes obtained by Transformation of Other Tessellating Shapes ) which can themselves be used to tessellate across a surface. Logo packages are also useful tools for exploring tessellation. When using Logo the onus is on the child: to consider the interior and exterior angles of the shapes they are using (thus developing their understanding of the mathematical reasons for tessellations being feasible or infeasible), to identify what constitutes the repeating unit for their tessellation, and to recognise how the repeating unit is replicated within that tessellation. The Mathematics of Tessellation ndrew Harris

24 The usual approach to tessellation with Logo is to encourage children to build up towards a completed tessellation as follows: 1 reate one or more sub-procedures which generate each of the shapes involved in the replicating unit; 2 Write a further sub-procedure which uses the shape sub-procedures to draw the unit of shape(s) that will be repeated; 3 reate a final procedure which: a) calls the sub-procedure that draws the repeating unit of shape(s); b) repositions the turtle to draw the next repeating unit of shape(s); c) repeats steps (a) and (b) until the desired surface area has been covered. Just as for any other area of mathematics in which IT is used, it is important to consider whether the use of IT is the most suitable tool/resource for achieving the desired learning or teaching objectives and thereby use IT as a means of teaching or learning only when it is appropriate. Resources The photocopiable sheets of tiling mats which follow can be photocopied onto card or paper and cut up to provide a selection of shapes for tiling large areas. Typical activities for using these with children are: creating tessellations which show different repeating patterns. How many patterns are possible? creating tessellations with closed or open patterns creating tessellations so that the pattern has the maximum number of regions possible is it possible to make pattern, make a larger version of the pattern which encloses the first, and an even larger version enclosing that... and so on? make a design with as many triangles as possible. create a pattern and investigate what happens if you slide one row of mats sideways How many mats are needed to make the smallest possible square? nd how many are required for a slightly larger square? nd the next square? creating tessellations which spell particular letters of the alphabet, words or numbers creating tessellations which have different kinds of symmetry (reflective, rotational, translational) design your own tiling mat by drawing or using the computer. How versatile is it? Use of these tiling mats adds an additional dimension to tessellation activities since, in addition to considering the position of each regular polygon within the tessellation, the children also have to think about the orientation of each tile and its contribution towards the pattern created as well. Tiling mats are also good for encouraging collaborative work and for developing the use and understanding of mathematical language. They also have uses in the teaching of area: for example, estimating and calculating how many tiling mats are needed to cover a given area (table-top, home corner, hall floor etc.). The Mathematics of Tessellation ndrew Harris

25 Hexagonal Tiling Mats The Mathematics of Tessellation ndrew Harris

26 Square Tiling Mats - 1 The Mathematics of Tessellation ndrew Harris

27 Square Tiling Mats - 2 The Mathematics of Tessellation ndrew Harris

28 Square Tiling Mats - 3 The Mathematics of Tessellation ndrew Harris

29 Useful References: Tessellation Deboys, M Lines of Development in Primary Mathematics lackstaff Press & Pitt, E. Wells, D 1991 The Penguin Dictionary of urious and Interesting Penguin Geometry Some Web site URLs The Mathematics of Tessellation ndrew Harris