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?
The cube's symmetry and special viewing angle make all projected edges appear equal.
Step 1 — Orienting the Cube
Let us imagine a cube balanced on one of its corners. This corner is the lowest point. The corner farthest from it is the highest point. The imaginary line connecting these two corners is called the main diagonal. This main diagonal is now perfectly vertical.
Step 2 — Understanding the Diagram's View
The diagram shows a special way of drawing 3D objects. This drawing method is called an isometric projection. In this projection, the cube is rotated in a specific way. The three main directions of the cube are equally tilted towards you. This makes them all appear shorter by the same amount.
Step 3 — Cube Edges and Projection
A cube has 12 edges. Let the actual length of each edge be . Each of these 12 edges is parallel to one of the cube's three main directions. These are the directions that are equally tilted in the projection.
Step 4 — Conclusion
Because all cube edges have the same actual length . And they are all foreshortened by the same amount in this view. So, all projected edges in the diagram appear to have equal length.
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?