Isometric projection

The idea

An isometric ('equal measure') projection of a cube is one in which the projections of all its edges are equal in length, obtained by balancing the cube on a vertex; it is a regular hexagon.

A cube on its corner

Look at a cube straight at a face: you see just a square. Now balance the cube on one corner and look straight down from above.

What shape do you see?

• A square
• A regular hexagon
• A triangle

Make the edges equal

A glass cube hovers above the floor. Its edges have three colours, one for each direction. The coloured lines on the floor are the projections of the edges. Each bar shows how long an edge of that colour is in the projection. A real edge is 1.00.

Equal measure

An isometric projection of a cube is one in which the projections of all its edges are equal in length. It is got by balancing the cube on a corner (a vertex), and it is a regular hexagon. Isometric means equal measure in Greek.

Six edges of the cube make the outline. The other six run from the centre to the corners of the hexagon, two by two along three straight lines. If the cube is made of glass, all the edges can be seen. This is the basis of isometric drawing.

Notes

An isometric projection of a cube is one in which the projections of all its edges are equal in length. It is got by balancing the cube on a vertex, and it is a regular hexagon.

Check yourself

Which picture shows the isometric projection of a cube, with all the edges equal?

In the isometric projection of a cube, every edge is 6 cm long on paper. What is the perimeter of the regular hexagon outline?

Answer: 36 cm

The outline is a regular hexagon with 6 sides, and each side is an edge of the cube, 6 cm long. 6 × 6 = 36 cm.

How must a cube be placed so that its projection on the floor is isometric?

A cube stands flat on a face, and Ravi looks straight down at it. He sees a square and says, "This is the isometric projection." Why is he wrong?

• Picture A. A is a square, got by looking straight at a face. The edges going up project to points, so the edges are not equal.
• Picture B. B is a hexagon, but its sides are not all equal: the cube is not balanced on a corner.
• Picture C — correct. Yes! C is a regular hexagon: the cube is balanced on a corner and all the edges are equal.
• Standing flat on one of its faces. Then you see a square: the edges going up project to points, so the three directions do not show equal lengths.
• Balanced on one of its corners — correct. Yes! Balanced on a corner, the three directions of edges lean the same way, so all the edges project to the same length.
• Resting on one of its edges. Resting on an edge, one direction of edges shows 1.00 and the other two show only 0.71. They are not equal.
• The edges going up project to points, while the other edges keep their full length, so the lengths are not all equal. — correct. Yes! In an isometric projection the projections of all the edges must be equal.
• A square cannot be the projection of a cube.. It can: a cube standing on a face has a square projection. It is just not the isometric one.
• A cube has to be made of glass to have an isometric projection.. The material does not matter. A glass cube just lets us see all the edges.