Drawing on an isometric grid

The idea

An isometric grid, the plane tiled with hexagons, is used to draw solids edge by edge, with the vertical direction for the height axis, the two slanted directions for length and depth, and equal unit steps along each.

Which way is up?

You want to draw a tower of three cubes on isometric paper. The paper has lines in three directions: straight up and down, and two slanted ones.

Which direction on the paper shows the height of the tower?

• Straight up and down
• Slanting to the right
• Slanting to the left

Read a drawing

This box is drawn on isometric paper. Its edges run in three directions. Tap an edge to see its axis and how many units long it is.

Draw four cubes

Tetris has five shapes, each made of four squares. The first one is a row of four cubes joined face to face. Put it along the depth axis: 1 unit long, 4 units deep, 1 unit high.

Edge by edge on the grid

On an isometric grid, a solid is drawn edge by edge. The vertical direction on the paper is the height axis of the solid, and the two slanted directions are its length and depth axes. A step of one unit is the same length along all three.

Why does it work? Parallel lines project to parallel lines, so the three families of parallel edges of the solid become three families of parallel lines on the paper. And the projection is isometric, so one unit along the height, the length or the depth is drawn equally long.

The grid comes from tiling the plane with hexagons, each one the picture of a cube.

Notes

On an isometric grid, the plane tiled with hexagons, a solid is drawn edge by edge. Vertical is the height axis, the two slanted directions are the length and depth axes, and a unit step is the same length along each.

Check yourself

A row of four cubes lies along the length axis. Which drawing shows it?

This cuboid is drawn on isometric paper. Its outline is a hexagon. How many units long is the whole outline?

Answer: 14 units

Going round the hexagon: 4 + 1 + 2 + 4 + 1 + 2 = 14 units. The length, the height and the depth each appear twice.

The edges of a solid along its height are all parallel. What do they look like on the isometric paper?

On an isometric grid, one unit going up is 2 cm on the paper. An edge of a solid is 5 units along the depth axis. How long is it on the paper?

Answer: 10 cm

A unit step is the same length along all three axes, so a step along the depth axis is also 2 cm. 5 × 2 = 10 cm.

• Drawing A. A goes up to the left, along the depth axis. The length axis goes up to the right.
• Drawing B — correct. Yes! The length axis slants up to the right, and the row is 4 units long along it.
• Drawing C. C goes straight up, along the height axis. It is a tower, not a row along the length.
• Lines that meet at a point. Parallel lines project to parallel lines, so they never meet.
• Parallel horizontal lines. The grid has no horizontal direction: its lines go straight up, or slanted. Height is the vertical one.
• Parallel vertical lines — correct. Yes! The three families of parallel edges become three families of parallel lines, and the height is the vertical one.