Isometric projection: the orientation that keeps all edges equal

0:00Stand a solid in the light and it throws a shadow on the floor. The shadow is a flattening: three directions go in, two come out, so something is always lost. The loss is uneven. Turn a box one way and its length comes out full size while its depth comes out half. Turn it another way and the depth vanishes altogether. That unevenness is the real damage. A drawing that shortens one direction more than another says one part of the solid is bigger than another when the solid says they are the same.

0:38So the question is not how to stop a shadow shortening things. It is whether the solid can be held so that the shortening is fair. Take a cube and ask a smaller question first. What shapes can its shadow be? Hold a face flat to the floor and the shadow is a square. The height is gone, and the top face lands exactly on the bottom one. Now tip the cube about one edge. The shadow stretches into a four-cornered shape whose two pairs of sides no longer match: one pair is the shadow of an edge, the other of a diagonal across a face.

1:20Tip it every which way, in no particular direction, and the shadow opens out into a six-sided shape - but a lopsided one. Its sides come out three different lengths. Three different lengths on the page for twelve identical edges on the solid. That is the lie in picture form. Which points at what a fair shadow would have to be. Every edge of a cube is the same length. So a fair shadow is one where every edge comes out the same length on the page. All twelve.

1:57That is a demand, not a description, and it might have no answer. There is a word for a shadow that meets it. Isometric - from the Greek for equal measure. The name is not a label for a style of drawing. It is the whole statement: the measures are equal. And equal does not mean unchanged. A flattening never lengthens anything, and unless an edge lies flat it shortens it. In a fair shadow every edge is shorter than the edge it came from. What makes it fair is that they all shorten by the same amount.

2:36So do not expect the picture to be life size. Expect it to be honest about proportion. Here is the orientation, and it is easier to describe than to guess. Set the cube on a face. Tip it onto an edge. Keep going, until it stands on a single corner with the opposite corner directly above it. Now let it drop its shadow straight down. Watch the shadow on the way. Square, then a squashed four-sided shape, then, at the moment of balance, a six-sided figure that looks suspiciously regular.

3:15Suspiciously is not good enough. The claim is that all twelve edges now measure the same on the floor, and that has to be argued. Look at the line the cube is balanced along: from the corner on the floor to the corner in the air. Turn the cube about that line, a third of a full turn. The cube lands exactly on itself. Every corner has gone to where another corner was, every face to where another face was.

3:47But the three edges meeting at the bottom corner have swapped places: the first to where the second was, the second to where the third was, the third back to the first. And the line you turned about is straight up and down, so the direction of the light did not move, and the shadow did not move either. Put those together. The shadow is the same shadow, and the first edge is now lying where the second one was. So their shadows are the same length, and the second's and the third's too.

4:22Three edges, forced equal, by a turn that changed nothing. Every other edge of the cube runs parallel to one of those three, and a flattening treats parallel segments of equal length alike. So all twelve are equal. No measuring. The answer came out of the symmetry. Now the outline. Why six sides? Look down that balanced line at the cube. Of its six faces, three lean towards you and three lean away.

4:54Every edge sits between exactly two faces, and most edges have both of theirs leaning the same way. The edges that make the outline are the ones caught between the two camps: a face towards you on one side, a face away on the other. Count them and there are six. Six edges on the boundary, so six sides to the shape - and each is an edge of the cube, so all six come out the same length.

5:26It is tempting to call it regular. Do not. Equal sides do not force equal angles. Here is a six-sided figure with all six sides exactly the same length, whose corners are not all the same. Some are blunter than others. It is equilateral and it is not regular. So equal sides are half the argument and no more. The other half is the same third of a turn. It carries the shadow onto itself, and it moves each corner of the hexagon to the next one round. A corner cannot be blunter than its neighbour if a single turn takes it there.

6:08Equal sides from the equal edges, equal angles from the turn. Now it is regular. There is more in the picture than the outline, and it starts with something strange. The corner on the floor and the corner in the air land on the same point - the middle of the hexagon - because the line joining them is exactly the direction being thrown away. So eight corners arrive at seven points. Six round the rim and two stacked in the middle.

6:41Three edges meet at the bottom corner and three at the top, and each of those six runs from a rim corner to the middle. Six spokes. On a solid cube you see three of them - the near corner's - and the far corner's three are hidden inside, so they are drawn broken. If the cube were glass, nothing would be hidden and all six would be drawn solid.

7:07And here is a check that the shape really is regular: the spoke and the side come out the same length, which is the mark of a regular hexagon. It was not arranged. It fell out. One hexagon is a drawing. A field of them is a tool. Slide a copy of the hexagon against one of the sides. It fits, sharing that side exactly: two corners and no more.

7:36Do it on each of the six sides and six copies close the ring around the first, none overlapping, none leaving a gap. Be careful which slide you make. Push the copy by a spoke and it also meets the first at two corners - but with a third corner between them, so it lies across the first, not beside it. Made properly, the ring keeps going outward and covers the whole page, and all those shared spokes and sides leave behind a mesh of small triangles: three families of parallel lines. That is the paper engineers draw on.

8:17The mesh measures along three directions, and those three are the cube's own. Length, depth and height. Four of the cube's twelve edges run along each one. On the page they come out the same length, and a third of a turn apart - the same third of a turn that started all this. So the grid is not decoration. It is a ruler in three directions at once, and the same ruler for each, so you can count along one direction and along the other two with the same eye.

8:53So what did this buy? A single straight-on view keeps two directions and destroys the third, so a box one deep and a box five deep give the very same picture. Balanced on its corner they do not. All three directions survive at one scale, so a size can be read off the drawing in any of them. That is a lot. Now be exact about what it is not.

9:20It is still a flattening. Two points a step apart along the direction being thrown away still land on the same spot - the two balancing corners are exactly such a pair. So the picture will not tell you which of them is nearer. A small crack - and the one that drawings which cannot exist are built out of.