Drawing solids on an isometric grid

0:00Four squares, joined edge to edge. Lay them out every way you can and you get five: the straight one, the square, the L, the S and the T - the five anyone who has played a falling-blocks game knows by sight. If you may not turn a piece at all, there are nineteen. If you may turn it on the table but not lift it, seven - the S and its mirror image being two pieces. If you may flip it over, five.

0:29Same shapes, three answers. Say what counts as the same, then say how many. Now make them out of cubes, and try to draw them. Freehand, a cube is hard. On the right paper it is not - and the right paper is a mesh of small triangles: three families of parallel lines, one straight up and down and one leaning each way. Those three directions are the solid's own three. Up the page is the solid's height; the two leaning ones are its length and its depth.

1:05Fix that once and do not swap it halfway through - swapping it is the commonest way these drawings go wrong. And what the paper does is arithmetic, not art. Take the three numbers that say where a cube is, subtract the height from each of the other two, and you have where it lands on the page. Three numbers in, two out. Start with one cube. One step along each of the three directions from a corner, and the outline closes by itself: six sides, each a single grid step.

1:43Three short lines inside, meeting at the middle, and it is done - it reads as a cube because three faces are turned towards you. Now a cube twice as big. You do not draw it again. Every edge that was one step is two steps: the same six-sided outline with every side doubled, and the three lines inside two steps each. That is the promise of the paper: a bigger solid is not a harder drawing, it is a longer count.

2:18Four cubes in a row. Two ways to get it onto the page, and they are not equally good. The first is to draw one cube, then the next beside it, then the next. It works, but each new cube covers lines you have already drawn, and those have to come out. With nothing to rub with, draw faintly and go over the surviving lines afterwards. The second is to count the edges before drawing any. Four steps along the length, one along the depth, one along the height - then only the lines you want.

2:55The difference is not neatness. The first is a drawing you correct; the second is an instruction list, and a list can be checked before a line goes down. Which lines are the ones that come out? A single cube has six faces and you see three. Every edge belonging only to the other three is one you do not draw. Push four cubes together and it is the same counting. Twenty-four faces to begin with; six get glued in pairs and vanish, leaving eighteen outside; of those, nine are turned towards you and nine away.

3:36Half the surface, exactly - which is why so many lines have to come out if you draw cube by cube. And a face can be lost the other way too: standing directly behind another cube along your line of sight. The same row of four can lie along any of the three directions: down to one side, down to the other, or standing up. Three drawings, and not one line of any of them falls where a line of another does.

4:07But every count is identical: four cubes, eighteen faces outside, nine turned towards you. The picture depends on how you lay it down. The counting does not. So why does this paper work at all? Two facts, held together. The first: parallel stays parallel. Edges of the solid that run the same way arrive on the page running the same way, and edges that do not, do not - so every edge pointing one way is one family of lines.

4:43The second: every unit is one step. This way of flattening treats the three directions alike, so one unit along any of them comes out the same length as one along either other. Either alone is not enough. Parallel without equal steps and you could not count; equal steps without parallel and you could not tell which direction you were counting along. Put them together and drawing stops being freehand and becomes arithmetic.

5:17Back to the four pieces, as cubes now. Are five still all of them? No. Five was the answer for shapes lying flat. Cubes can leave the plane. Stand the fourth cube on one end of the L rather than beside it: a new shape, in two forms that are mirror images. Or take the corner and let the fourth rise out of the bend. Count them and there are eight, if a shape and its mirror image are two. Seven, if they are one. Three more than five, or two.

5:51And the difference is exactly one pair: of the eight, six look the same in a mirror and two do not. The same lesson as at the start. State the convention, then state the number. Three shapes to draw, and nothing to do but count. An L standing up: four cubes, two long, one deep, three high. A T lying low: four cubes, three long, two deep, one high. A staircase: six cubes, three long, one deep, three high.

6:24For each edge the only decision is which of the three directions it runs along, and whether with it or against it. Answer that for every edge and the drawing appears. Which makes it sound as though the paper carries everything. It does not. Here is one cube. Put a second directly behind it - one step along the length, one along the depth, one along the height, all at once. That is the direction the flattening throws away.

6:58The second lands exactly on top of the first. Not near it. On it. Two cubes and one cube, and the drawing is the same drawing, line for line. A drawing on this paper does not say how deep the solid goes. A block of cubes with a route marked on it, running round and coming back to where it started - gaining height at every turn. Every arrow sits on a real face of a real cube, so any short stretch of it could be built this afternoon. The circuit could not.

7:34You do not need cubes to see why. If a route comes back to where it started, the heights it gained must cancel the heights it lost. If it only ever gains, they cannot. Every route of up to six steps in which every step goes up was searched, and not one closes. Let the steps go down as well and the same search finds closed routes at once - so it is the climbing that is impossible, not the search that is blind.

8:06So what closes? The drawing. The route moves the same number of steps along all three directions, and that is exactly what the paper cannot show. Which brings us to a triangle: three straight arms of cubes, meeting at three corners, drawn on this same paper. Ring each corner in turn. Each is two arms meeting at a square angle - a join real cubes make every day. Take any two of the three arms and you can build them: a row, a turn, another row, no cube asked to be in two places.

8:45Now follow all three. Four steps along the length, four along the depth, four along the height - and where you end up is four steps along the direction the paper throws away. So the object has not closed. It has gone twelve steps further back, four for each arm. The drawing has closed, because those twelve steps land on nothing. The last cube is drawn where the first is while being far behind it, and nothing on the page can say so - no shrinking with distance, no converging lines.

9:22Each corner asks you to assume its two arms are at the same depth. Each time the assumption is harmless. Three together are not. One property, two consequences. The flattening treats the three directions fairly, which is why you can count on it; it still loses a direction, which is why it can be fooled. The useful grid and the impossible picture are one fact, seen twice.