Idea Seeds #10 – Visualization, Geometry and the Platonic Solids
Natures Wonders and Geometry
The ancients were very aware that if they didn’t get to grips with some basic ‘geometry’ much of the world would remain a mystery to them. In previous articles I wrote that ‘Visualization’ and ‘Geometry’ go hand-in-hand and that they are both critically important ‘idea-seeds’. I learnt this and that the very act of making simple models using things like drinking straws, cardboard, polystyrene, pins, paper clips, tooth picks, fruit, vegetables and your imagination help improve your ‘visualization’ and importantly help you start understanding the ‘geometry’ that is inherent in every model, from my father. Then you too, as Edna Millay wrote in one of her sonnets about Euclid, ”can look on beauty bare” and make the links between ‘nature’s wonders’ and ‘geometry’.
A knife, a Fork, a Bottle and a Cork
By making a model of the solution to the following problem you will see just how helpful a model can be in aiding ‘visualization’. All you require is a cork from a wine-bottle, a sharp knife, some thin cardboard and a pencil.
Visualise the Task
The task is to ‘visualize’ the shape of a single solid object that can pass through each of the three holes shown in the adjacent template and also completely block each of the holes at some point as it is pushed through them. The sides of the square and the base and height of the triangle are all equal to the diameter of the circle. Think about it before reading further.
Making the Model
To make the model, start with the template. Place the cork on a piece of thin cardboard and mark round it with the pencil. Cut out the circular disc and make sure the cork can pass through it. As you check this, you will see that the cork does block the hole in the template completely.
Next mark and cut out a square in the template. When you look at the cylindrical cork from the side it is rectangular in shape. Cut a piece off the cork to leave a cylinder with height equal to its diameter. It will block the square hole completely when it is half way through the square hole in the template. Next draw the triangle on the template. I suggest you start by drawing another square and then mark the mid point of the upper side. Join this point to each of the bottom corners of the square and you will have a triangle with the base and height equal to the diameter of the cork. Cut out the triangular piece. Next use the pencil and draw a line across one of the circular ends of the cork so it divides it exactly in half. The length of this line will then be equal to the diameter. Set the knife on this line and cut along it and downwards at an angle that ensures the knife reaches the bottom but does not remove any part of the circular base. If you have difficulty ‘visualizing’ this then insert the cork into the square again with the line at right-angles to the template. You should then be able to see what needs to be cut away to allow it to pass through the triangular hole.
Bananas and the Art of Circles
You will now have a model of a solid that fulfils all the requirements specified in the problem statement. When you study the model you will see where you sliced the bits off to make the triangular shape, that the cuts produced two semi-ellipses (semi = half). If you had no previous knowledge of what an angled cut through a cylinder produces then, without a model, you will have had great difficulty in ‘visualizing’ the solution.
When you next decide to eat a banana, before doing so, do a bit of experimenting on it. After peeling it cut through the banana at right angles to its axis. The cut will produce two pieces with identical circular ends. Make another cut but this time at a slightly smaller angle to the axis. The cut will produce identically shaped elliptical ends. The shape of the ellipses will change as the angle of the cut to the axis is decreased. The ‘minor axis’ which is fixed by the diameter of the banana will remain the same but the ‘major axis’ will get longer. Every time you do exercises like this, don’t forget to investigate the ‘limiting’ conditions. Clearly going from a circular to an elliptical shape as the angle to the axis of cut is decreased means you are moving away from a ‘limit’. A circle is in fact a special case of an ellipse. As the two foci of an ellipse are brought closer together the shape of the ellipse becomes more circular. When the foci meet the ellipse becomes perfectly circular; something for you to ponder on when you study the elliptical path of a planet on its path round the sun shown in the diagram below in the paragraph titled Johannes Kepler.
Euclid wrote ‘Elements’, a "Holy Little Geometry Book"
Euclid of Alexandria is considered to be the ‘Father of geometry’. He collected two and a half centuries of discoveries made by earlier mathematicians such as Thales of Miletus (624 – 546 BC), Pythagoras of Samos (About 570 – About 495 BC) and Plato (423 – 348 BC) and then wrote a set of books titled ‘Elements’. They were used continuously as textbooks for more than 2000 years, right into the early part of the 20th century. His systematic and logical approach in laying out each theorem together with a rigorous proof ‘demonstrating’ how it linked to a small set of axioms, was his great gift to the world. Abraham Lincoln kept a copy of ‘Elements’ in his saddlebag to study because he said: "You never can make a lawyer if you do not understand what ‘demonstrate’ means.” Albert Einstein said that two gifts that had influenced him most when he was a boy were a magnetic compass and a copy of ‘Elements’ which he referred to as his "holy little geometry book".
Euclid in ‘Elements’ covered: geometry, perspective, conic sections, proportion, number theory, spherical geometry, the Platonic solids and a lot more. Sadly little is known about him other than he lived in Alexandria around 300 BC.
The symmetry and simple beauty of the five ‘Platonic solids’ have been studied and marvelled over for thousands of year. Each has: Faces that are all the same shape and size; Edges that are all the same length; Interior angles between faces that are all equal; And, when put into a sphere of the right size, all the points touch the surface of the sphere. An octahedron inside a cube is shown in the adjacent diagram. Each of the points on the octahedron is touching one of the faces of the cube. This configuration is called a 'dual'. A cube with its eight points can fit inside an octahedron with each point touching one of its faces. You can carry on doing this an infinte number of times. Google 'Sacred geometry' to see some of the beautiful things the ancients created.
The Big Five
There are only five solids with these characteristics:
- ‘Tetrahedron’ with four equilateral triangles;
- ‘Cube’ with six squares;
- ‘Octahedron’ with eight equilateral triangles;
- ’Dodecahedron’ with twelve regular pentagons;
- ‘Icosahedron’ with twenty equilateral triangles.
The ancients linked four of these to the four elements of Greek science − earth, air, fire, and water and the fifth to the universe. Even the great Johannes Kepler (1571 – 1630) was enchanted by them and thought he had found the answer to how the planets were positioned round the sun in nested Platonic shapes. A sketch of his model is shown in the diagram below.
Today of course Johannes Kepler is remembered for his three planetary laws. His first law states that the orbit of a planet is elliptical with the Sun positioned at one of the foci. The second law states that a line drawn from the planet to the sun sweeps out equal areas during the same passage of time. This means that when the planet is close to the focus it has to travel faster than when it is further away to ensure the area swept out is the same. This is one of the reasons why the rotation of planets round the sun makes it difficult to use them for accurate time keeping. In his third law he shows mathematically how the time a planet takes to travel around the Sun increases significantly the farther away it is from the sun. For example, the Earth takes a year while Pluto takes two hundred and forty seven years. More about these important laws when ‘time’ is explored in more detail.
Toothpicks and Jellytots
Start collecting the model making bits and pieces mentioned in the article and store them in a box so, when in the future you need to quickly and with minimal fuss make a simple model, material will be at hand. In the adjacent photograph you will see three models made in a few minutes using toothpicks and jelly-tots.