© ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes [email protected] for permissions
Mathematical Activity Tiles
- the nextgeneration
GEOMETRY WITH CUT MATS
a booklet about the 2D and 3 D geometrical possibilities opened up by cutting regular polygonal tiles
Paul Gailiunas Adrian Pinel
Academic copyright permission does NOT extend to publishing on Internet. Provide link ONLY © ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes [email protected] for permissions
MATHEMATICAL ACTIVITY TILES
The first MATs were designed by Adrian Pinel and produced by ATM in 1980. Since then the range has expanded so that among those now available are the seven regular polygons below.
regular hexagon equilateral triangle
regular octagon
regular pentagon
regular dodecagon 2 Academic copyright permission does NOT extend to publishing on Internet. Provide link ONLY © ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes [email protected] for permissions
From April 1996, A TM have available the following next generation of MA Ts, which are based upon cuts of the regularpentagon:
acute angled 'pentagon triangle' 'pentagon rhombus'
obtuse angled 'pentagon triangle' 'cocked-hat'pentagon
MATs began with the equilateral property that allowed different MATs to be edge matched. With two of these new MATs this propertyremains, but with the new triangles, there is a different edge length. This is in golden proportionto theoriginal. Through this, many exciting new possibilities for 2D and 3D exploration of space are opened up.
Not all of the ideas and explorations in this book are restricted to cuts of the pentagonal MATs. Some require the cutting of the larger MATs, fromregular hexagon to regular dodecagon.
3 Academic copyright permission does NOT extend to publishing on Internet. Provide link ONLY © ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes [email protected] for permissions
PENTAGON TRIANGLES
,:\
A pentagon can be cut into triangles in an obvious way to give two obtuse pentagon triangles and a single acute pentagon triangle.
Golden Ratio The number that is equal to its reciprocal plus 1 : 1 if>=-+1 if> Solving this equation gives: 1 + if>= VS '" 1.618 2
Both of these triangles have the lengths of their sides in the golden ratio. It is quite easy to see this by comparing the I +-1 lengths of corresponding sides of similar triangles.
Each triangle is the gnomon for the other.
4 Academic copyright permission does NOT extend to publishing on Internet. Provide link ONLY © ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes [email protected] for permissions
Gnomon
A piece which can be added to a shape to produce a similar shape is called a gnomon. A golden rectangle has its sides in the golden ratio. Adding a suitably sized square produces a larger golden rectangle, so a square is the gnomon of a golden rectangle.
Gnomons can be cut off repeatedly so that, for instance, the acute triangle can be made from a series of obtuse triangles plus an acute triangle that can be as small you like. Alternatively gnomons can be added to cover as large an area as you like.
In both triangles the gnomons converge to a point that lies on a medianl. Median
A line joining a vertex of a triangle to the mid-point of the opposite side. The three medians intersect at the centre of mass of the triangle at a point one third up each one.
You can keep on doing this, so that a large triangle can be tiled with a mixture of acute and obtuse pentagon triangles that are as small as you like. Alternatively an area as large as you like can be tiled using a mixture of the two triangles. Since each cut produces a triangle of each type there will be equal numbers in the resulting tiling pattern.
1 A proofin the case of the acute triangle is given in Does the Golden Spiral Exist, and ifnot, Where is its Centre, A.L.Loeb and W.Varney in Spiral Symmetry, I.Hargittai and C.A.Pickover (eds.), World Scientific,I992
5 Academic copyright permission does NOT extend to publishing on Internet. Provide link ONLY