Visual Thinking with Triangles

We use triangles to visualize certain types of mathematical and non-mathematical relations and concepts. Unlike 2d and 3d visualizations, triangles aren’t mathematically coherent in any intuitive way. So let’s try and figure out the logic of triangular representations of ideas, and why we use triangles so extensively but not other shapes. Here are two common examples:

The first is a representation of a commonly recognized set of tradeoffs among quality, speed and cost of a service, with the simplest interpretation being “you can have any two.” I have no idea who invented this one, but you’ll see signs whimsically asserting that tradeoff at some pizza places. You’ll find one exploration in The Age of Speed by Vince Poscente, which I reviewed earlier.

The second is attributed to the physicist D. T. Spreng, and I found it Impossibility: The Limits of Science and the Science of Limits by John Barrow. The idea there is that to accomplish any given task, you can use a mix of three “pure” capabilities: unlimited knowledge and no time or energy, unlimited time, but zero knowledge or energy, or unlimited energy and zero time and knowledge. You can also have various mixes of the three. The pure cases are, approximately, the archetypes of muscular caveman, great philosopher and immortal lazy idiot.

Here are some more examples that I won’t try to draw:

1. Consider an idea guys toss around half-seriously: that women are either smart or beautiful, never both. A friend of mine, worried by the sexist nature of the proposition, but trying to get to any truth it might have, once proposed an extension, “The product of a woman’s brains, beauty and availability is bounded by a constant.” Can a 2d-triangle represent this 3d constraint approximately?
2. Consider color spaces. Color is infinite-dimensional as a physical phenomenon (a reflectance spectrum), but given the the biology of vision, 3d representations work well. It turns out that triangles are among the most intuitive representations of color. In Adobe Photoshop Elements for instance, you select a hue in a hue circle, and then pick a point inside a triangle to determine lightness/darkness and saturation. Other color interfaces use triangles in different ways. If this example doesn’t make sense to you, don’t worry. I only know this color-science-101 stuff because I work in the print industry.
3. There is also a simpler use of triangles in color science to represent the notions of additive and subtractive primaries (red-green-blue and cyan-magenta-yellow respectively) and how they relate.

What is the Mathematical Structure Here?

The pizza triangle works like this. If you view each side as a binary on/off constraint, then making the vertices represent the design space represents the idea that only two of the constraints can be active at any given time. If you want to interpret a point on the side or in the interior, things get a little more convoluted, because the variables are now [0, 1] bounded-range variables. You have to interpret each side as representing the associated quantity at its maximum value, and the opposite vertex as the zero. One end of the slider itself slides on a conceptually orthogonal edge, and the scale stretches but always represents the same interval. This means the red point is maximum cheapness, and a mix of high speed and low, but not inedible quality. Interior points extend the idea further.

The Spreng triangle works differently. Unlike in the pizza triangle, the vertex captures the idea that only 1 variable can be maximized at any given time. Sides represent binary tradeoffs with the third variable set to zero. The interior is a ternary tradeoff space. The relation between each edge and its opposite side is reversed, since the vertex is the maximum and the opposite edge is the zero. Note that in this particular example, vertices represent unachievable “infinity” points which we arbitrarily normalize to 1 (i.e. infinite knowledge=1).

So, to generalize, if you have three (normalized) bounded quantities and the boundaries of the design space include the points (1, 0, 0), (0, 1, 0) and (0, 0, 1), you have a Spreng-like, or S-class triangular design space. The Pizza-like, or P-class, is bounded by (1, 1, 0), (1, 0, 1) and (0, 1, 1).

A Challenge to Readers

Clearly, since a 3d space is being constrained to 2d, the actual constraint space is some sort of manifold that hits the boundary points, but I can’t imagine the shape.

I am lazy but curious, so here is a challenge to readers: the first person to send me computer-plots of the manifolds and associated 3d solids representing the convex hulls of the actual constraint spaces of classes P and S, will win a used book from my collection. The only thing I can guess about the shape is that P-class will be more convex-looking, since the centroid represents the more “bulgy” (2/3, 2/3, 2/3), which is not included in the S-class, where the centroid is (1/3, 1/3, 1/3). While it is hard to compare two manifolds in 3d, I’ll guess that in some sense, with the same limit points, the P class is a larger space (maybe in terms of surface area/Lebesgue measure?).

Also, is there a deeper interpretation of the fact that this particular pair of 2d manifolds within a 3d constraint space turns out to be so useful?

I’ll post a follow-up blog with any contributions I get, or you can post on your own blog and I’ll link.

The Other Examples

Of the other examples, 2 is easy to interpret, because it is derived from an actual mathematical object. In the Adobe example, the triangle is a conservative approximation (or possibly a non-linear exact map, I don’t know for sure) of a vertical slice through a complex polyhedron called a gamut.

Example 1 is harder. Both class S and class P can be used (for example, the class-S would be to make the vertices pure beauty, brains or availability), but it is not clear which is better or if the triangle is a useful approximation at all. Maybe in this case, we have a true 3d continuum as the meaningful constraint space (remember the constraint is xyz<=constant, and if you add a positive-only constraint, you get your true constraint solid).

Example 3, while the most familiar, is actually the hardest, since the additive/subtractive color triangles represent operation spaces rather than constraint spaces. Each point in the triangle represents the result of a combination operation.

Why Only Triangles?

I can only guess that quadrilaterals and higher-order polygons are simply too hard to use productively. Remember that the number 3 has an interesting property: the number of unique pairs of 3 variables is also 3. With 4 variables, you get 6 unique pairs and you have to draw diagonals, and it is not clear that pair-wise analyses are as useful in higher dimensions.

Other Uses of Triangles

For completeness, let me note some other ways triangles appear in visual thinking.

• They are often used with an up-is-more gravitational-orientational metaphor to represent things like reporting hierarchies, levels of decision-making and the like.
• Then, you also have more mathematical ideas like the triangle inequality, which might have some interesting visual interpretations. Someday I’ll mull that.
• Then, there is their rigidity, which made them so useful to Buckminster Fuller.
• Finally, being one of the two simplest polygons (the other being the circle, with an infinite number of sides), triangles are often used in conceptual analysis.

Someday, I’ll write a more thorough exploratory piece on triangles. Right now, I am stuck in Rochester airport mulling the pizza triangle as applied to budget airline travel on holidays (today being Thanksgiving).