Let's rotate a cube mentally!

Geometry Level 2

Consider a cube with a thin rope attached to one of its vertices. The other endpoint of this rope is fixed to the bottom of a lightbulb which is hanging on the ceiling.

What is the shape of the shadow thrown on the floor by the cube?

Note: The only light source in the room is the lightbulb.

Bonus: If the light bulb is infinitely far away, what can we say about the shape?

Circle Triangle Square Rhombus, but not square Parallelogram, but not rhombus or square Hexagon Octagon

3 solutions

Dan Ley
Apr 3, 2017

This is the view of a cube from above, which is the shape of the shadow for light bulbs infinitely far away (where the light rays are effectively parallel).

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Is the shadow still a hexagon if the string is attached to the cube's edge rather than the vertex? Does it matter where on the edge the string is attached?
Is the shadow still a hexagon if the string is attached to the cube's face? Does it matter where on the face the string is attached?

Attached to the midpoint of the edge, I imagine the shadow would be a rectangular shape. Then I would guess that as you approach the vertex, the shadow approaches a hexagon.

Again, attach it to the midpoint of the face and the shadow would be square, but in other positions we may have stranger shapes... right now, my Rubik's cube is not handy so I can't tell for sure!

Dan Ley - 4 years, 2 months ago

What does it mean to "approach a hexagon". IE Is there a placement of the string that will produce a pentagon?

Calvin Lin Staff - 4 years, 2 months ago

If there were points attached to the midpoints of the left and right sides of the rectangle, it would be as if these points were being pulled apart to form a hexagon. (I think)

Dan Ley - 4 years, 2 months ago

The vertex is only one point, but an edge is a line segment

Mike Blane - 4 years, 2 months ago

it doesnt matter where on the edge the string is attached, since wherever it's attached on the edge or the face respectively, the shadow won't change its shape. when it's attached to the face, it is in the form of a square, while on the edge it is a rectangle.

Qgqyqikavgwhya Aggahhqvagqhqb - 4 years, 2 months ago

Your comments assume that we're attaching to the center of the edge / face. What if it's not the very center? Does the shadow change?

In particular, we know that as we move from the middle of the edge to the vertex, it goes from a rectangle (as you said) to a hexagon (answer to this problem). How does that change happen? Is it always a rectangle on the edge, and then suddenly a hexagon on the vertex?

Calvin Lin Staff - 4 years, 2 months ago

It's rectangle only at the middle of the edge. Once it's off-centre, they'll be irregular hexagonal shadow shapes and keep getting more regular until it reaches the vertex.

Saya Suka - 2 months, 3 weeks ago

Marta Reece
Mar 20, 2017

The hexagon will be close to regular if the light bulb is at a sufficient distance. Otherwise the closer trio of vertices makes a larger equilateral triangle than the more distant one.

Thanks. I've edited the options to "Hexagon" instead.

Calvin Lin Staff - 4 years, 2 months ago

It is better that a rubik's cube on your table.

Kelvin Hong - 4 years, 2 months ago

Steve Powersuit
Apr 3, 2017

Just look at the corner of a cube and count the edges: 6. Bonus: If the light was infinitely far away, the shadow would be the same apparent size as the box (2 dimensionally speaking).

If the light-bulb is infinitely far away the light will never reach the box

Gabriel Schmidt - 4 years, 2 months ago

Exactly. The lines of sight from your eyes are just like the rays of light from the lightbulb. So this provides an easy way to work it out.

Stewart Gordon - 4 years, 2 months ago

Gabriel stole my joke.

Richard Desper - 4 years, 2 months ago

Let's rotate a cube mentally!

3 solutions

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