Constructing a solid in your head from a description

0:00Everything in the next ten minutes is meant to be done in your head. Not drawn. Held, and then acted on. That sounds like a talent some people have and others do not. It is an operation, and you can be shown it and then get better at it. Tesla, the engineer, said he built his machines in imagination first. Altered them there. Ran them there. Only then made them out of metal.

0:28You are not being asked for that today. You are being asked to hold one shape and do one thing to it, and then look at what you are left with. Start with the smallest version of the task. Picture a word. Chalk. Five letters, sitting in front of you. Now read it off backwards. Not the sound of it backwards, the letters. K, L, A, H, C. Notice what you just did. You held something still and ran an operation over it.

1:00Do it twice and you are back where you started. And some words come back unchanged. Level, read backwards, is level. That is the whole method. Hold, act, look at the result. Now a shape. Picture a square. On each side, mark the middle. Then slice off each corner, cutting straight from one middle mark to the next. Four cuts. What is left? Most people expect eight sides, because four cuts on a four-sided shape sounds like eight.

1:34It is four. The two cuts meeting at a corner arrive at the same two marks, so no piece of the original side survives between them. What is left is a square, standing on its corner, with its four sides equal and its four angles right. And it is smaller. Take the original side as one; the new side, squared, is a half. Now look at what you cut away. Four triangles.

2:04Each one is right angled, with two short sides of a half and two angles of forty five degrees. Here is the part worth stopping on. Slide those four triangles together and they make a square. Not any square. That square. The one still sitting on the board. The arithmetic says so before the sliding does. Each corner is an eighth of what you started with, so the four of them are a half.

2:32And a half is exactly what is left behind. One line finishes it. The long side of a cut corner is the side of the square left standing, so the pieces fit that outline and no other. Change the marks and the answer changes with them. Picture an equilateral triangle. Mark each side at the thirds, so two marks per side, then cut the corners away as far in as those marks.

3:03Three cuts this time. What is left? Six sides, because each cut takes one corner and leaves the middle third of the two sides beside it. And it is regular. All six sides are a third of the original side. All six angles are a hundred and twenty degrees. A regular hexagon, out of a triangle, from one instruction. Which makes the next one look easy, and it is not.

3:33Same marks, on a square. Thirds on every side, corners cut away as far as the marks. Eight sides. Everybody gets that far. And all eight angles are equal, at a hundred and thirty five degrees. So it is a regular octagon. That is what nearly everyone says next, and it is wrong. Measure the sides. Four of them are the middle thirds of the original sides. Four of them are the cuts.

4:04Squared, those two lengths are a ninth and two ninths. The cut is the longer one, by a factor of the root of two. Eight equal angles and two different lengths. Equal angles do not make a shape regular. So why did the triangle work and the square not? It is nothing to do with those two shapes. It is the corner. Cut a corner at the thirds and two things appear. The cut across the corner, and the piece of side left beside it.

4:37The piece left is always a third. The cut depends on how wide the corner was. At sixty degrees, the two come out equal. At ninety, the cut is longer. At a hundred and twenty, longer still. Take the three regular shapes with those three corners: the triangle, the square and the hexagon. Cut all three at the thirds. Exactly one of them gives a regular figure, and it is the triangle. That is a fact about sixty degrees, not about triangles.

5:14Now lift the whole thing off the page. When you see a solid object, what actually reaches you is not the object. It is the shape the object presents from where you happen to be standing. Its profile. And the boundary of that profile is its outline. There is a cartoon version of this that gets it exactly right. Something charges through a wall and leaves a hole shaped like itself.

5:42The hole is the outline, made physical. It is a record of one profile from one viewpoint, punched into brick. Now walk round to the side and charge again. Different hole. Same solid. So the solid does not have an outline. It has one per viewpoint. Take a cylinder, a tin of soup. Look at it end on and the outline is a circle. Look from the side and it is a rectangle.

6:13One object, two outlines that share nothing at all. A cube is the other extreme. Square, square and square, whichever of the three ways you look. But even that is not the reassurance it seems. Stretch the cube into a cuboid and two of those three become rectangles. And there is exactly one solid here whose outline really does describe it. A ball. A ball is a circle from everywhere. It is the one shape with nothing to hide from any direction.

6:49Now run it the other way. I will name an outline. You find a solid and a place to stand. A square. A cube, seen face on, will do it. So will a tin whose height happens to equal its width, seen from the side. A circle. A ball, from anywhere. A tin, end on. A cone, from directly underneath. A triangle. A cone from the side. A prism with triangular ends, seen end on. A pyramid on a triangle.

7:21Notice what happened to the question. Every single answer came in twos and threes. Of the fourteen solids on this board, eight show a circle from some direction. Four show a square. Six show a triangle. So a circle on a wall tells you almost nothing. It rules some things out, and that is all it does. Harder version. Two outlines at once, from two viewpoints, on one solid. A rectangle and a circle. The tin again.

7:55A circle and a triangle. A cone: triangle from the side, circle from below. A rectangle and a triangle. The prism with triangular ends. A wedge sliced off a box does it too. A trapezium and a circle. A cone with its top cut off. A bucket. A pentagon and a rectangle. A prism whose ends have five sides. And now a warning, because this is where a good guess goes wrong.

8:26A cone standing on half a ball feels like another answer to the circle and triangle one. From below, yes, a circle. From the side it is a triangle sitting on a curve, which is not a triangle. It fails, and only measuring it says so. Last question, and it is the one the whole thing was built for. Are those answers unique? Take the cone again. A taller, narrower cone gives the same two outlines. That is a rescaling, so perhaps it does not count.

9:03So here is one that is not. An open cup, hollow, with a cone's slope. From the side, a triangle. From below, a circle, because the outline is the boundary of what you see and the hollow never reaches it. Every viewpoint gives the same answer for the cup as for the solid cone. They are not the same object. That is what an outline is. Evidence about a solid, not a description of one.

9:33One solid gives you many outlines, and one outline is given by many solids, and neither of those directions is a fluke. Which is exactly why there is a whole subject about drawing solids properly, and why it needs more than one view.