Knot Theory Pretzels

The classic pretzel shape involves a simple twist and is attributed to a number of sources, most with religious connotations. One popular story is that pretzels were invented around 610 A.D. by a baker–monk who wanted to find a use for leftover scraps of dough. He twisted a length of dough and formed the pretzel’s iconic shape, which was meant to resemble crossed arms. The pretzel’s three holes are said to represent the Holy Trinity.

Although classic pretzels are what we see most often, it’s just one of many ways the soft, clay-like dough can be shaped. From a bow tie to a dagger-like corkscrew, the shapes are limited only by your imagination. You’ll find a number of different step-by-step pretzel shaping guides in chapter 25 of Modernist Bread.

Shaping pretzels provides a tangible example of a branch of mathematics known as knot theory; it explores which knots are truly different from one another and which can be manipulated until one is equivalent to another. These determinations are far from obvious—in one case, it took nearly 100 years before one knot believed to be distinct was shown to be the same as another. This relatively young field has proved useful as well as elegant, with applications in biology (think of knotted DNA strands), chemistry (which involves molecular knots), and physics (for the study of quantum mechanics), as well as expanding into theories about the creation and structure of the universe.

Mathematically speaking, a knot is a closed loop that can’t be undone by untangling its threads: a circle, known as a trivial knot, is the simplest form. Working from that basic foundation, we shaped pretzels into some of the simpler mathematical knots (such as the elementary trefoil), sometimes weaving two or three pieces of dough together, depending on the complexity of the pattern. In scientific theory, the shapes would be considered infinite, with no start or finish. In our earthbound dough version, we tucked the ends of each pretzel underneath…and then added salt.

We based our pretzel knots on illustrations of mathematical knots. While there are many possibilities for shaping pretzels according to knot theory, we found that the simpler knots showcased the patterns the best because the strands didn’t proof and bake together.

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