Question 9
Construct a model of a cube and use your hands to keep it balanced on one corner vertex. Can you try to understand why all the projected edges have equal length?
- When a cube is balanced on one of its corner vertices, its body (main) diagonal is oriented vertically.
- In this viewing angle (known as an isometric projection), the three mutually perpendicular axes of the cube are equally inclined toward the observer.
- Because all edges of the cube are of equal actual length and are tilted at the exact same angle to the viewing plane, they all foreshorten by the same scale factor, appearing equal in the 2D projection.
Step 1 · Orientation of the Balanced Cube
When the cube is balanced on one vertex, the line connecting this lowest vertex to the diagonally opposite highest vertex forms the vertical main diagonal.
Step 2 · Isometric Projection and Edge Lengths
In this balanced orientation (an isometric projection), the three principal axes of the cube make equal angles with the projection plane.
Let the actual length of each edge of the cube be .
Every one of the edges of the cube is parallel to one of these three axes. Since all three axes are equally tilted relative to the viewer, each edge is foreshortened by the exact same proportion, resulting in equal projected lengths.
All projected edges have equal length because the cube's three principal axes are equally inclined to the projection plane in an isometric view, foreshortening every edge of length by the exact same factor.
- True Length vs. Projected Length: Confusing the actual 3D edge length () with its 2D projected length. The projected length is shorter than , but all projected edges remain equal to each other due to symmetry.
- Orientation Dependency: Assuming edges look equal in all orientations. This equal foreshortening happens specifically because the cube is aligned along its main diagonal (isometric projection).
More questions in A
Build it in Your Imagination
We will start this section by practising visualisation. For each prompt, feel free to talk to your partner, gesture, draw it in the air — but do not actually draw on paper!
- Picture your name, then read off the letters backwards. Make sure to do this by sight, not by sound — really see your name! Now try with your friend's name.
Cut off the four corners of an imaginary square, with each cut going between midpoints of adjacent edges. What shape is left over? How can you reassemble the four corners to make another square?
Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What shape do you get?
Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left?
Net of a sphere? Experiment and see if you can make a paper cutout that can perfectly wrap around a ball without leaving any wrinkles, gaps or overlaps.
Place an object in front of a plane, such as a wall of your room. Shine a torch light on the object in a direction perpendicular to the wall.
What do you see?
Observe what happens to the size of the shadow as you vary the distance between your torch and your object.
Context: Observe what happens to the size of the shadow as you vary the distance between your torch and your object.
Q. Why does this happen?
Construct a model of a cube and use your hands to keep it balanced on one corner vertex. Can you try to understand why all the projected edges have equal length?