The Brownie Simplex: A brief expose on the scientific method mixed with design thinking as applied to the active and ongoing area of brownie research.

Written by Drew J. Lipman, Ph.D. at Hypergiant

There is this almost institutionalized segregation separating art and science. On one hand we have nose-in-the-air “artists,” many of whom are teaching art in elementary and middle school or are selling their wares in the craft shows dressed in hemp clothing. On the other hand we have dour, seemingly humorless “STEM people” who work at convoluted problems that rarely make any sense to people not in their particular subfield. These views give rise to this seemingly impenetrable divide separating “design thinking” of art from “analytical thinking” of the STEM fields. Yet, on the third hand, many of the most successful people, in an art field or a STEM field, have a blend of both.

Ranging from Bertold Brecht (who wrote the Threepenny Opera, a personal favorite) who said “Art and science work in quite different ways: agreed. But, bad as it may sound, I have to admit that I cannot get along as an artist without the use of one or two sciences. In my view, the great and complicated things that go on in the world cannot be adequately recognized by people who do not use every possible aid to understanding.” to Werner Heisenberg (from whom the Heisenberg Uncertainty Principle is named) who said “Therefore, the two processes, that of science and that of art, are not very different. Both science and art form in the course of the centuries a human language by which we can speak about the more remote parts of reality, and the coherent sets of concepts as well as the different styles of art are different words or groups of words in this language.”

What is to be done? How are we bridging this divide? Only by accepting both types of thought into our everyday lives and using them in our everyday problems.

To counter this segregation, and to provide an amusing, sort of real world, example of how Design and Analytical Thinking can work and play off each other, I would like to talk about brownies. As my poor office mates can attest I have a deep seated interest in baking. Moreover, baking illustrates how these two types of thinking can interact, but I will do it in what is, perhaps, an unexpected manner. Consider an old recipe, like the ones my, and probably your, grandmother, or great-grandmother would have used. These often were nothing more than a list of ingredients, as the nature of the ingredients, the ratio of liquid to solids, determined the method of assembly. In that light, we can apply an analytical approach to recipes by parameterizing recipes onto a simplex. What is a simplex? A simplex is the generalization of a triangle. A one dimensional simplex is a line segment, two dimensional is a triangle, and three is a three sided pyramid. We will be using the Simplex defined by the points (e_1, e_2, … ,e_n) where e_i represents the point which is all zero except the i-th coordinate, which has a 1. One of the special properties of this simplex is that it can be written as the sum of any number of variables set equal to 1. For example, a three variable equation (x + y + z) = 1 is, in general, equal to an ’n’ number of variables, which would also equate to (x_1 + x_2 + … + x_n) = 1. OK, that is the last of the formulas you can wake up from your math fugue state.

Let’s talk about what these equations mean: they tell us that all the variables together give 1 and they are never negative. So, if each variable represents an ingredient then we never have a negative amount of an ingredient, and all the ingredients together give the entire brownie. Therefore, for any brownie recipe, where the ingredients are measured by their percent of the recipe, it is represented by a point in this simplex.

Now that we have a geometric representation of the space of brownie recipes, what do we want to do with it and how do we come up with the best brownie recipe in the world? We apply analytical thinking: we have a geometric space, we can define an objective function on the space, and then apply optimization techniques to produce a superior brownie recipe. Let’s walk through the steps: using a current recipe I make a batch of brownies. I bring the brownies to the office and force-feed them to my teammates and demand reports of the quality, in triplicate. Using these reports I produce a direction on the simplex that would potentially improve the brownie, I then adjust the recipe to reflect that and save it for the next occasion I am feeling like baking. So, where is the design thinking? Think about this, when soliciting from colleagues how they like your brownies they are not going to reply with “it needs more flour!” or “too much raspberry jam” they will be providing abstract sensory feedback “I prefer cakier brownies” or “this brownie is too fruity”. Only by applying design thinking, the artistic side, can you translate what these, often contradictory, opinions mean into a format, a terminology, that the analytical thinking, the science side, can interpret. The design thinking is fundamental to the entire operation. The design thinking is not something that gets slapped on top of an existing analytical method, it is fundamental to the entire operation. It is how we determine the correct improving direction, it is how we interpret everything.

Moreover, as anyone who deals with gradient based optimization will tell you, starting from a single point will not produce a global optimal solution, the best brownie recipe, just a local optimal the best one of a certain type. To fix this issue of finding yourself at a local, but not global, optimal brownie recipe there are several analytical methods that can be applied. The one I personally implement is to maintain several different “lines” of brownie recipes ready for iteration, and if two get too similar to each other I create a new one. So how do we create a new starting point for our explorations of the brownie simplex? We choose an area that we have not really explored too much and grab a (mostly random) point from that area. However, there is a design layer to this as well. Some areas can be ignored. Determining if they can be ignored is not analytic, it is sensory: “too much flour, this would taste like flour and nothing else.” Or knowing that two flavors do not really mix together, or knowing that this combination would produce something that is not pretty. Since people enjoy food, not just with their mouths but their eyes and noses all of these senses play together to determine where a good starting point is for a new line of brownie research.

Now, I can almost hear some of my readers muttering “isn’t this a little insane?” To which I reply: Of course it is, but that is why it is fun! In the words of Albert Einstein “We cannot solve our problems with the same thinking we used when we created them.” Analytical thoughts can only push things so far, design thoughts can only push things so far: only by using them together can we achieve more. In the case of the Stochastic Gradient Descent Algorithm applied to the Brownie Simplex, well, the results speak for themselves. This method has produced some wonderful, diverse, brownie recipes including Orange Almond Brownies, Espresso Balsamic Brownies, or Raspberry Jam Fudge Brownies. So to all of you who say “I can’t do art” I tell you it does not have to be an art form that anyone else appreciates or understands in order to be artistic, valuable, and fun. So long as you see the artistic side and are willing to use Design thinking you will achieve better results. To all of you who say “science has no place in art” I tell you that you are handicapping yourself by not using all the tools available to further your art, and by willfully ignoring the right tool to achieve the desired result because it does not fit into your narrow world view, you are dooming yourself to hemp shoes for life. So, with that final thought I am off to continue my explorations of the wonderful world of Brownies!