Wednesday, July 9, 2008

"Any Way You Slice It" - A Classic Cube Dissection Problem to the Nth!

The following series of questions was inspired by a recently released SAT question. The first two levels are appropriate for middle or secondary students. Level III requires more algebraic background or strong visualization skills. I could have attempted to include a graphic for some of this but I'll leave that to the experts out there!

LEVEL I
A cube is cut into 8 equal cubes by dividing each edge in half with three planes which are parallel to the faces of the original cube. Show that the total surface area of the 8 smaller cubes (when separated) is TWICE the surface area of the original cube.

Note: Most secondary students would attempt this algebraically or substitute particular values. To develop spatial sense, encourage them to find another solution, which is purely visual and elegant! Middle school students (or younger children) would greatly benefit from constructing a physical model of the cube from modeling clay (or something equivalent) and slicing it with appropriate tools. Better yet, one can avoid slicing by constructing the bigger cube from 8 smaller cubical blocks (There are many sets of plastic or wooden blocks available from catalogs).

LEVEL II
A cube is cut into 27 equal cubes by dividing each edge into 3 equal parts with planes parallel to the faces of the original cube. Show that the total surface area of the 27 cubes is THREE times the surface area of the original cube.

LEVEL III
Generalize the above relationships by dividing each edge of a cube into N equal parts with planes parallel to the faces of the original cube (N is an integer greater than 1). State a conclusion and explain! Again, try to find both an algebraic and a visual explanation.

9 comments:

Eric Jablow said...

I won't talk about the solution you're seeking, but I do see it. However, I would suggest that anyone teaching this immediate discuss the consequences of the square-cube law in engineering, medicine, athletics, architecture, chemistry, and other related subjects.

Why do grain silos blow up?

Why do mice and similar-sized animals have no problems falling from a height, but people do?

Why doesn't a person's size really matter in a marathon, but does in a sprint?

Why does one start with twigs when one builds a campfire?

The fastest horse can run a mile in just under two minutes. The fastest human can run a mile in 3:45. What makes "ride and tie" races plausible?

Why do the proportions of the human body change so radically between infancy and adulthood?

Paleontologists and medical examiners can estimate the size of a body quite well from just one thigh bone. How?

The answers come from the same cause.

Dave Marain said...

I like that real-world connection, Eric! Comparing the stress on a one sq in base supporting ONE cu. in to a one sq in base supporting TWO cu in when we double the dimensions is a powerful application for students. Thank you for reminding me of that. The square-cube law is so vital in geometry and other applications yet I do not believe it gets enough attention. Am I wrong?

PhD in Math? Pfft! What a waste! said...

Hi!

I love this problem! Part of the reason why I like it is because formal approach to it (i.e., say describing first the cube in 3-space algebraically and taking it from there, or something more sophisticated, such as projecting its faces first onto a 2-space and taking it from there) would be, to say the least, inconvenient. A more conceptual, rather informal approach is the best one to take here. This demonstrates the many faces of mathematical reasoning. Strict formalism rarely promotes creativity (in my opinion... although Euler, Hardy, Ramanujan and other die hard formalists would probably disagree).

Questions like these demonstrate an important concept: if one understands the "why," the "how" usually becomes easier.

Personally, when working on problems, I usually speculate, imagine, draw silly figures, ask silly questions and leave formal, technical details until after I've gotten the "feel" for the "why."

Now on with the solution. I'll do the most general case. I'll give a formal argument which hopefully is intuitive enough to illustrate the basic idea (and it is this idea that students should see and not the details of the argument).

Say we're given a cube C with faces F1, ..., F6. Say we dissect the cube using N planes per each face: P11, P12, ..., P1N, ..., P31, P32, ..., P3N; these planes need not be equidistant.

Let F be the set of faces of the cube and P be the set of dissecting planes. Now, each face of each of the smaller cubes lies in one element of F or P (and only one). However, each element of P contains TWO faces - namely those of adjacent cubes. Thus, the sum of the surface areas of each one of the smaller cubes is the sum of areas of planes in F, plus twice the sum of areas of planes in P (assuming each plane in P has the same area as the the faces of our original cube). So:

Total = F1 + ... + F6 + 2*(P11 + ... + P1N + ... + P31 + ... + P3N)

Assuming that all faces are of equal area:

Total = 6*F1 + 2*3*N*F1 = 6*F1 + 6*N*F1 = 6*F1*(N + 1).

Notice that 6*F1 = area of original cube.

So Total = (area of original cube)*(N + 1).

Again, I wouldn't expect students to give an argument like this, but I would expect them to see the "strategy" behind the proof above.

Humbly Yours,

W.N.Y.

PhD in Math? Pfft! What a waste! said...
This comment has been removed by the author.
Joshua Zucker said...

The usual variations of this problem are worth mentioning too:

If you paint the outside of the cube ... now you know that if it's an nxnxn cube that 1/n of the surfaces of the 1x1x1 cubes are painted, which is cool.

But that's an average -- some cubes have 0 surfaces painted, some have 1, some have 2, some have 3. How many of each?

And then the slightly more interesting variation -- most interesting, I think, when n = 3. How do you paint the 1x1x1 cubes such that they could be assembled into an nxnxn cube of any one of n colors?

Joshua Zucker said...

Oh yeah, and I imagine that the solution you're seeking goes something like "look at the top of the cube: you'll see that there are n layers, of which only the topmost layer is part of the surface area of the nxnxn cube, so 1/n of the top surfaces are ..."

Florian said...

Level 1:

The simpliest visual solution that I could find is:

When the 8 smaller cubes are assembled to form the original/big cube each cube has three sides that lie inside the big cube and three that lie on the surface of the big cube. Therefore the total surface of the smaller cubes has to be two times the surface of the big cube.

Level 2:

Finding the solution seems to be essentialy the same as for Level 3
so I ommit it here. Onle only needs to argue with n=3.

Level 3:

Let C be the cube.
Let a be the length of an edge of C.
The surface of a side of C is a^2.
The total surface of A of C is 6a^2.

The total area of the smaller cubes equals the surface area of C (=A) plus the sum of the surface areas of the smaller cubes inside C.

Each parallel plane that cuts C creates two new surfaces with area a^2.
If we cut C along each of its three axis (with equidistant parallel planes) n-times, the total sum of the newly created surfaces is 2a^2 * 3n = n*A.

Thus the total surface area of the smaller cubes is A + nA = (n+1)A.



I think this is a nice problem because it is very visualy (=easy to understand the problem) and requires not only visualisation but also some degree of abstraction to solve.

If you want to force students to think even further, try ommiting that the cutting planes are equidistant to each other.

Eric Jablow said...

Consider the three directions. The cuts in one direction do not contribute to the surfaces in any other direction. Now, making (n-1) cuts in a direction (dividing the cube into n parts) adds 2(n-1) faces in that direction, resulting in 2n faces in that direction. Do this in all three directions, and you can see that the surface area is multiplied by n.

Let L be the characteristic length in the following.

Why do grain silos blow up?

Dust builds up, and has a high surface area to volume ratio. The availability of the grain dust to react with oxygen is high.

Why do mice and similar-sized animals have no problems falling from a height, but people do?

The forces on a falling object are gravity, proportional to mass (L³), and air resistance, proportional to cross-sectional area (L²). The smaller the animal, the higher the second is in comparison to the first.

Why doesn't a person's size really matter in a marathon, but does in a sprint?

In a sprint, the force a person can put out is proportional to mass (L³), and the force against him is air resistance, proportional to cross-sectional area (L²).

In a longer race, the force a person can put out is limited to the amount of waste heat he can rid himself of, proportional to his surface area (L²). If someone tried to run a mile at sprint pace, he'd develop heat stroke.

Why does one start with twigs when one builds a campfire?

Twigs have a high surface area to volume ratio; it's easy to set them afire. Once they're hot enough, branches put amidst or atop them will burn; then logs put atop those will burn.

The fastest horse can run a mile in just under two minutes. The fastest human can run a mile in 3:45. What makes "ride and tie" races plausible?

In a "ride and tie" race, teams of two people and one horse compete. One person starts running; the other rides the horse and passes her. Whenever the rider sees fit, she dismounts, ties the horse to a convenient place, and runs further along the course. The original runner eventually reaches the horse, mounts, and rides along the course, passing the original rider.

This is a long race, so the limit on the horse's speed is surface area, proportional to (L²). The force against the horse is air resistance, also proportional to (L²). So, the explanation of the different running speeds of a man and a horse is that the body shapes and physiology lead to different proportionality constants, but this will be insensitive to distance traveled.

The runner catches up to the horse. The horse cools down, and is fit to ride again.

Why do the proportions of the human body change so radically between infancy and adulthood?

There are many reasons, but the important one is that the lungs must grow faster than the rest of the body in order to provide oxygen to it, and so the chest must grow faster than the rest of the body to hold the lungs. Oxygen transport across the lungs is proportional to the surface area, but mass is proportional to volume.

Paleontologists and medical examiners can estimate the size of a body quite well from just one thigh bone. How?

Different parts of a body have different characteristic growth rates. Furthermore, the examiner can determine the approximate age at death of the body the bone came from by looking at its structure. It's not quite like tree rings, but it's close.

jd2718 said...

I tried to extend this over here

Jonathan