Class 8 Mathematics Chapter 11: Exploring Some Geometric Themes

A shape made of itself

A fern is one of the loveliest examples of a fractal in nature. Its leaves are smaller copies of the whole fern, and the sub-leaves of those leaves are smaller copies again. A shape that behaves like this is called self-similar.

Where else it happens

Nature is full of the same trick. A tree trunk carries limbs, each limb carries branches, and the branches carry branchlets. Clouds, coastlines, mountains and lightning all show the same pattern repeating at smaller and smaller scales.

A fractal made on purpose

The Polish mathematician Sierpinski invented one. Take a square, cut it into 9 smaller squares, and remove the middle one. Then do exactly the same to each of the 8 squares that remain, and carry on for ever.

What the figure holds

At every step there are two things worth counting. First, the squares that remain, all of them the same size and getting smaller as the steps go on. Second, the square holes left where pieces were removed.

Naming the counts

Write R for the squares remaining and H for the holes, with the step number underneath. So R at step 0 is 1, since we start with one whole square, and H at step 0 is 0, since nothing has been removed yet.

The rule for the squares

Here is the step that turns a picture into arithmetic. Every square that remains at one step becomes 8 squares at the next, since it is cut into 9 and loses its middle. So R at step n plus 1 is 8 times R at step n.

Turning that into a formula

Follow it along. R at step 0 is 1, R at step 1 is 8 times 1, which is 8, and R at step 2 is 8 times 8, which is 8 squared. The pattern is clear: R at step n is 8 to the power n.

The rule for the holes

Holes behave differently. Every square remaining at one step leaves one new hole at the next, and every hole already there stays. So H at step n plus 1 is H at step n plus R at step n.

The holes, counted out

That gives 0 at step 0, then 1, then 1 plus 8, then 1 plus 8 plus 8 squared. Adding them up, the holes number 0, then 1, then 9, then 73. They pile up faster and faster because the squares feeding them are multiplying.

How much is left

Each step keeps 8 of every 9 parts, so the area surviving is 8 ninths of what it was. Starting from 1 square unit, step 1 leaves 8 ninths and step 2 leaves 64 over 81. The area shrinks towards nothing while the pattern goes on for ever.

The same idea on a triangle

Sierpinski built a second fractal the same way. Join the midpoints of an equilateral triangle, which cuts it into 4 identical equilateral triangles, and remove the middle one. Repeat on the 3 that remain. This is the Sierpinski gasket.

Counting the gasket

The arithmetic runs just as before, with 3 in place of 8. Triangles remaining go 1, 3, 9, 27, so the count is 3 to the power n. The holes go 0, 1, 4, 13, and the area left is 3 quarters to the power n.

A fractal that grows outwards

The Koch snowflake, described by the Swedish mathematician Von Koch in 1904, starts from an equilateral triangle. Every side is cut into 3 equal parts, an equilateral triangle is raised on the middle part, and that middle part is then removed.

Counting its sides

The effect is that each side is replaced by a bump made of 4 shorter sides. So the sides go 3, then 12, then 48, then 192, which is 3 times 4 to the power n.

Counting its perimeter

Each new side is a third as long as the one it replaced. Starting from a side of 1 unit, the perimeter is 3, then 4, then 16 thirds, then 64 ninths. The boundary grows at every step and never stops growing.

Fractals in art

People have built fractals by hand for a very long time. The Kandariya Mahadev Temple at Khajuraho, finished around 1025 CE, is made of smaller copies of its own shape, which carry smaller copies again. Temples at Madurai, Hampi and Varanasi show the same thing.

Further afield

Nigerian Fulani wedding blankets often carry diamonds holding smaller diamonds. And the Dutch artist M. C. Escher built the idea into prints such as Smaller and Smaller, where one pattern of lizards repeats at ever smaller scales.

Scenario:

Three pupils are planning a display. The first is drawing a Sierpinski carpet. She wants the squares and the holes for step 3. The second is drawing a Sierpinski gasket. He claims that since each triangle becomes 4, the count must be 4 to the power n. The third is drawing a Koch snowflake from a triangle of side 1 unit. She says the border of it must get shorter at every step, since every new side is only a third as long as the one it replaced. All three want their answers checked before any paint is opened.

Analysis:

One needs two different rules at once, one has counted the pieces before the removal, and one has forgotten how many pieces there are.

Title:

Measure a growing border

Safety:

Nothing here is risky, but a sharp pencil makes the small sides much easier.

What You Notice:

Each perimeter comes out as 4 thirds of the last, so they run 27, then 36, then 48 centimetres. The figure never leaves the page, and its border grows without limit.

Some of the oldest fractal art in the world stands in India. The Kandariya Mahadev Temple at Khajuraho, finished around 1025 CE, is built from smaller copies of its own shape, which carry smaller copies again.

Von Koch described his snowflake in 1904. Its boundary grows longer at every step and never stops growing, though the whole figure stays inside a small patch of paper.

Count what one step does — Find what a single piece becomes at the next step. Multiply by that each time, and the formula follows.

What we actually see

When we look at a solid object, we are not seeing all of it. We are seeing its outline from one particular viewpoint, which is called its profile. Move around the object and that outline can change completely.

A familiar version of the idea

Cartoons use it all the time. Somebody charges through a wall, and the gap left behind has the shape of their own silhouette. A shadow does the same thing, and so does a photograph.

One outline, many solids

Try it the other way round. What solid gives a circular outline? A sphere does, but so does a cylinder seen end-on, and so does a cone seen from directly above. One outline never names one solid.

Two outlines narrow it down

Pairs of outlines are far more useful. A solid that looks rectangular from the side and circular from the end is a cylinder. One that looks triangular from the side and circular from above is a cone. A triangular prism gives a rectangle and a triangle.

Naming the parts of a solid

For solids with flat surfaces, three words do most of the work. Faces are the flat surfaces that form the boundary. Edges are the line segments where faces meet. Vertices are the points where edges meet.

The cube, counted

A cube or cuboid has 6 faces and 8 vertices, with 12 edges running between them. It is worth counting those on a real box once, because the three numbers are easy to confuse and hard to forget after one careful count.

Prisms

In a prism, one pair of opposite faces is a pair of congruent polygons, with edges running between their matching corners. Every other face is a parallelogram. The name follows the polygon, so there are triangular prisms, pentagonal prisms and so on.

Counting any prism

Take a prism whose polygons have n sides. There are the two polygons plus n parallelograms, so n plus 2 faces. Each polygon has n edges and n more join them, giving 3n edges. And there are n vertices at each end, so 2n in all.

Pyramids

A pyramid has a polygonal base and a single point outside it, with edges joining that point to every vertex of the base. Again the name follows the base: triangular, square, pentagonal. A triangular pyramid is also called a tetrahedron.

Counting any pyramid

For a base with n sides there are the base and n triangles, so n plus 1 faces. The base has n edges and n more rise to the apex, giving 2n edges. And the vertices are the n of the base plus the apex, so n plus 1.

A check worth doing

For a 10-sided prism those rules give 12 faces, 30 edges and 20 vertices; for a 10-sided pyramid, 11 faces, 20 edges and 11 vertices. In both, faces plus vertices less edges comes to 2, which is a useful way to catch a slip.

Flattening a solid

Many solids can be made from paper or cardboard by folding. The trick is to draw a flat shape that folds up into the solid, and such a shape is called a net. Equivalently, a net is what you get by unfolding a solid onto a plane.

What a net is not

To build a real box you also want flaps to glue the faces together, and sturdy enough material to stand up. Neither belongs to the net. A net is only the shape the solid unfolds into, with nothing extra.

How many nets a cube has

There is no single net of a cube. Counting two as the same when one is a turn or a flip of the other, a cube has exactly 11 nets. An octahedron, made by joining two square pyramids base to base, also has 11.

Other counts

A regular tetrahedron, whose faces are all equilateral triangles, has only 2 nets. At the other extreme, a dodecahedron, whose 12 faces are all pentagons, has 43,380 of them, which mathematicians have counted exactly.

Unrolling a cylinder

A cylinder is not made of flat faces, but it still unfolds. Take off the two circles and cut the curved surface along its height, and it opens into a rectangle. One side of that rectangle is the height; the other is the circumference of a circular face.

Unrolling a cone

Slit a cone along a line from its tip to the base and unroll it. Every point of the base circle was the same distance from the tip, so the unrolled boundary is an arc of a circle centred where the tip was. The net is that sector, plus the base circle.

The one that will not flatten

Try the same with a ball and it fails. No paper cutout wraps a sphere without wrinkles, gaps or overlaps, so a sphere has no net. That is exactly why every flat map of the world distorts something.

Scenario:

A stage crew is cutting card for four props. The first is a hexagonal prism tower, and they need its faces, edges and vertices for the painting plan. The second is a square pyramid, and a crew member says it must have the same counts as the prism because both are five-sided in name. The third is a cylinder, and they want its net. The fourth is a large ball, and someone has been asked to draw its net so the card can be cut in one piece.

Analysis:

One needs the prism rules, one confuses the two solids, one needs an unrolling, and one is asking for something that does not exist.

Title:

Unroll a tin and a cone

Safety:

Watch for sharp edges on the opened tin, and ask an adult before using one.

What You Notice:

The rectangle’s sides come out as the height and the distance round the rim. The opened cone is a sector of a circle, because every point of the rim was the same distance from the tip.

A cube has exactly 11 nets, counting two as the same when one is a turn or a flip of the other. An octahedron also has 11, and a dodecahedron has 43,380.

A sphere has no net at all. No flat cutout wraps a ball without wrinkles, gaps or overlaps, which is why every world map distorts something.

Faces flat, edges meet, vertices point — A face is a flat surface. An edge is where two faces meet. A vertex is where edges meet.

A hungry ant

Across a flat sheet, two points are joined most economically by the straight line between them. Now put an ant on one face of a cuboid and a laddu on another. What is the shortest route, if the ant may walk only on the surface?

Why guessing is not enough

A route may look short and still not be. There are infinitely many paths across a box, so no amount of looking settles it. Something has to turn the question into one we can already answer.

The idea

Draw the net of the box. Any path along the surface can be traced onto the net without changing its length, and any path on the net folds back onto the box, again without changing its length. Lengths survive the unfolding.

What that buys us

So the problem becomes a flat one. The shortest path on the net is the straight line between the two points, and that line folds back into the shortest path on the box. A route that is not straight on the net is therefore not the shortest.

Testing a guess

This gives a test for any proposed route. Unfold the box, trace the route onto the net, and look. If it comes out straight, it is the shortest. If it bends, a shorter route exists.

A complication

A cuboid can be unfolded in several different ways, and the straight line is not the same in each. Worse, in some unfoldings the straight line between the ant and the laddu runs outside the net altogether.

What that means

A line that leaves the net matches no path on the box at all, so it must be discarded. The way a box is opened out matters, and a careful solver has to try the different unfoldings rather than trust the first.

Measuring one

Take a box 30 cm long, 12 cm wide and 6 cm high, with the ant on one face and the laddu stuck to the back. One unfolding puts them at the opposite corners of a right triangle whose legs are 24 cm and 32 cm.

Finishing the calculation

Now the Baudhayana theorem does the rest. The square of the distance is 24 squared plus 32 squared, which is 576 plus 1024, or 1600. So the distance is root 1600, which is exactly 40 cm. A question about walking round corners has become a single right triangle, which is the whole payoff of flattening the problem first.

Flattening a solid another way

A net is one way of putting a solid on paper, but it can be hard to picture the solid from it. A second way is to record where the solid’s points land when they are projected onto a plane.

The projection of a point

Take a point P and a plane M. Drop a line from P meeting the plane at O. If that line is perpendicular to the plane, meaning it makes a right angle with every line of the plane through O, then O is called the projection of P.

The projection of a whole object

Do that to every point of an object and the results together form the projection of the object. It is closely related to the profile: if the object passed perpendicularly through a wall, the hole would have the shape of the projection.

What happens to a length

A line and its projection are rarely the same length. Drawing the right triangle shows that the projected length is never greater than the true length. The two are equal in exactly one case: when the line runs parallel to the plane.

One projection is not enough

Different objects can cast the very same projection. Several different lines, and several different cuboids, give identical results. So a single projection cannot be relied on to say what the object was.

Three planes instead of one

The remedy is to take three projections at once, on three mutually perpendicular planes: a vertical plane in front of the object, a horizontal plane below it and a side plane beside it.

Naming the three

On the side plane the projection is the side view; on the horizontal plane it is the top view; and on the vertical plane it is the front view. Together they make the profiles of the earlier section exact.

Shadows do it for you

Shine a torch at an object straight at a wall and the shadow looks much like the projection, though it may be stretched. Move the torch further back and the stretching shrinks. The Sun is the far-off torch this needs, so perpendicular sunlight casts shadows that are projections.

A fact worth keeping

Cut out a parallelogram and watch its shadow in such sunlight. However the cutout is turned, the shadow stays a parallelogram, because parallel lines always project to parallel lines. That one fact makes drawing projections far easier.

Scenario:

A workshop has three questions on the board. A spider sits on one face of a crate and a crumb lies on another. An apprentice has drawn a route over two faces and called it the shortest, because it looks direct. A second crate unfolds so that the straight line between spider and crumb passes clean outside the net. Nobody knows what to make of that. Finally a bracket is being drawn. Its single front view has been handed over as a complete description of the part.

Analysis:

One claim rests on looking, one is about a line that corresponds to nothing, and one treats a single projection as enough.

Title:

Make shadows into projections

Safety:

Never shine a torch into anyone’s eyes, and do not look directly at the Sun.

What You Notice:

The further back the torch goes, the less the shadow is stretched, until it matches the projection. In sunlight the shadow of a parallelogram stays a parallelogram however it is turned, because parallel lines project to parallel lines.

Sunlight is the projector this chapter needs. When sunlight falls perpendicular to a wall, the shadow it casts on that wall cannot be told apart from a true projection.

Whatever you do to a parallelogram cutout, its shadow in perpendicular sunlight stays a parallelogram. Parallel lines always project to parallel lines.

Flatten it, then rule a line — Unfold the box, join the two points with a ruler, and fold it back. The line was the shortest path all along.