Here is another solution to the cube doubling problem, which didn't make it into my recent video. It uses origami, which means that six-year-old me would have been thrilled:

• First, fold a piece of paper into three equal sections (this can be done with a precise algorithm).
• Unfold, then bring the bottom left corner to the other side, such that the corner lies on the right edge and the left endpoint of the lower fold lies on the upper fold.

• The corner splits the right edge into two lengths. The ratio of the top length to the bottom length is the desired ratio, cube root of 2 to 1.

It’s as simple as that.

Proving this is straightforward too, but the algebra gets a little messy. It would be a great exercise for a student.

The key is to use the diagram's similar right triangles.

(Here is a written proof, and here it is in video form)

Why Didn’t the Greeks Just Do It This way?

Well, technology was a major barrier. Paper wasn’t invented until around 100 CE, and papyrus simply cannot accommodate origami style folding.

I would also argue that this solution doesn't solve the problem in the way that the Greeks wanted anyway.

You can't start with a unit length (such as your 1 x 1 x 1 sized altar to Apollo) and then construct the cube root of 2 from that length. Instead, you construct the desired ratio between the two sections of the paper and you can only determine the original "unit length" after the construction is complete.

But comparing this origami construction to the Greek approaches can give us some interesting insights.

Why the Greeks Loved the Ruler and Compass

For Greek mathematicians like Euclid, the straightedge and compass give a physical manifestation of a clear set of starting assumptions. So following the rules of constructions automatically means that you are adhering to the axioms.

These rules are as follows:

1. We can generate points at any arbitrary locations.
2. We can join any two points into a line segment and extend that line segment indefinitely.
3. We can find the intersection point of any two non-parallel lines.
4. We can create a circle with a given center point and a second point that forms a radius.
5. The intersections between any generated lines or circles can be designated as new points, which can then be used in new constructions.

Once a point is created, its position on the page is fixed.

Distances are not measured, nor are things ever "wiggled" around to make them fit in a specified way.

(Euclid does crucially use an idea known as superposition in The Elements, which arguably presents inconsistencies with what I’ve just described, but let’s gloss over that for now)

How Can Origami Do What Ruler and Compass Can't?

The axioms implied by paper folding are similar to ruler and compass, but it has key differences which give this method its added power:

1. Given two points, there is a unique fold that passes through both (forming a line)
2. Given two points, a unique fold places p1 on top of p2, forming the perpendicular bisector of the line segment connecting the points.
3. Given two lines, a unique fold places one line on top of the other (essentially constructing an angle bisector if the lines are not parallel)
4. Given a point and a line, there is a fold that passes through the point and is perpendicular to the line (found by making a fold that goes through the point and folds the line onto itself)
5. Given two points and a line, there exists a fold passing through one point which puts the second point somewhere on the line.
6. Given two points and two lines, there exists a fold which can put the first point onto the first line and the second point onto the second line.

Read more about the "origami axioms" as described by Humiaki Huzita, in this article, which includes some important further notes at the end addressing issues of rigor.

Axiom #6 is the operation which leads to the cube doubling construction.

It turns out that the capabilities of this fold are equivalent to solving the cube problem using neusis (a sliding, measured ruler), as well as Philon’s construction, which involves “jiggling” a measured ruler back and forth until the setup forms congruent line segments as shown.

These methods allow the construction of solutions to third degree polynomials, while the straightedge and compass can only find solutions to quadratic equations. The added capabilities of these methods come at the expense of the strictness and simplicity of straightedge and compass.

What does this tell us about the Greeks?

The sheer abundance of proposed solutions to the cube doubling problem using approaches that go beyond straightedge and compass seems to suggest that the Greeks were not sitting around for centuries, trying and failing to solve the problem with straightedge and compass, as some versions of the story might imply. It seems likely that they indeed had some understanding that cube roots occupied a different category than "constructible magnitudes."

But what I find most remarkable about looking at these many solutions to the cube doubling problem is the sense that I'm seeing a snapshot of an ongoing ancient debate.

While a student today experiences geometry as being fixed and unchangeable, much of what we see as obvious was once anything but.

What I'm currently working on

My next video will be about Al-Khwarizmi, considered the father of algebra and the namesake of “algorithm.”

For me, the early stage of these videos -- facing the blank page -- is often the hardest part.

It can take a few failed drafts before I figure out what my video is really about. In this case, I'm finding that it will take some narrative finesse to clearly articulate Al-Khwarizmi's unique contribution to the math that he inherited from the Greeks, Indians, and Babylonians.

If Al-Khwarizmi, the Islamic Golden Age, and the origins of algebra is a topic of interest or expertise to any of you, I'd love to hear from you!

Until next time

Ben