THERE’S NO DANCE WITHOUT MUSIC

To dance, music is essential. Dance steps match the rhythm of the music, and that rhythm consists of measures, which are the mathematical way of marking the duration of sounds. Just look at how they are represented: with fractions. We have the 2/4, the 4/4… The denominator is always a power of 2, and the higher the value, the shorter the duration of each note. The numerator tells us how many notes make up each measure.

In the music of Basque dances, there are two uncommon measures that stand out: the 5/8 time signature found in the Zortziko, and the 6/8 time signature of the Ezpata Dantza, where it alternates with 3/4 time.

DANCING AT THE COMMAND OF GEOMETRY

In choreographed dances like ballet or Basque dances, dancers arrange themselves in a more or less orderly way in the performance space. When we imagine lines connecting their positions, geometric shapes start to form.

In Basque dances, circles appear frequently. For example, in the Ezpata Dantza of Xemein, twelve dancers (dantzaris in Basque) form a circle, with another dantzari standing in the center. In the Zinta Dantza, eight dantzaris form a circle around a central pole.

Polygons are also common. Often, dancers stand in two parallel lines, creating a rectangle. This happens in Basque dances, but here we’re a bit more creative and can even see pentagons, like in Brokel Dantza, where one dantzari leads eight others arranged in two parallel lines of four. In Txotxongilo, we even see a hexagon: four dancers line up centrally, while the others are arranged two per side, forming a hexagon when we connect the outermost dancers’ positions.

PERMUTING AND COMBINING DANCERS

Combinatorics is a branch of mathematics that studies the possible ways to order and group elements in a set. For this, each element is assigned a number: 1, 2, 3…

A permutation is a reordering of a set of elements, represented by listing those elements in the new order. Let’s look at an example: if we have the set {1, 2, 3}, all possible ways to arrange these numbers are: (1 2 3), (1 3 2), (2 1 3), (2 3 1), (3 1 2), and (3 2 1). Each of these is a permutation, and with three elements, we see there are six distinct ones.

Permutations appear in the Zortzinango, part of the Dantzari Dantza, a traditional dance from Durangaldea involving eight dancers. These eight dancers stand in pairs, one behind the other in two parallel lines, and they change positions throughout the dance. Let’s assign numbers to the dancers’ positions, starting with the leftmost in the front pair and moving left to right and front to back. With the dance’s first movements, the two front pairs of dancers, who form a square’s vertices, switch places with the dancer at their opposite corner of that square. The same happens with the dancers in the rear. This permutation is represented as (4 3 2 1 8 7 6 5) because the dancer in the first position has moved to the fourth, the dancer in the second position has moved to the third, and so on. Then each dancer swaps with the one beside them. This permutation is (2 1 4 3 6 5 8 7). Here’s a challenge: can you figure out the next permutations associated with the dancers’ movements? You’ll be able to watch the Dantzari Dantza in the link, try to write out those permutations!

Combinatorics also studies the different ways of choosing a specific number of elements from a set. For instance, from the set {1, 2, 3, 4}, there are four ways to select three elements: {1,2,3}, {1,2,4}, {1,3,4}, and {2,3,4}. If, instead of four elements, we have an arbitrary number n, and we want groups of k elements, we have combinations of n elements taken kat a time. In the Binango, another part of the Dantzari Dantza, only two of the eight dancers perform simultaneously, but which two are dancing changes throughout. Here we have combinations of eight dancers taken two at a time. Something similar happens in the Launango, where four dancers perform simultaneously, giving us combinations of eight dancers taken four at a time.

BRAIDS, MATHEMATICS, AND DANCE

To braid, we divide the hair into three strands and weave them over and under in an orderly way. There’s even a branch of advanced algebra that studies and generalizes braiding: braid theory. In braid theory, we have a set number of strings hanging from an imaginary bar and study the patterns formed when they cross over or under one another. These crossings are treated as elements in a set, forming an algebraic structure known as the “braid group.” If that imaginary bar is a circle instead of a straight line, the last strand can cross with the first, forming what’s known as a “circular braid group.”

Two dances incorporate braiding with colored ribbons instead of hair strands. In Iruleak, eight pairs of dancers hold ribbons and alternately pass them over or under the pair next to them during the dance. If these ribbons were attached to a bar behind the dancers, we could see how a braid in the colors of the Basque flag is formed. This braid can be described mathematically using braid group notation. In Zinta Dantza, each dancer holds a ribbon hanging from a central pole, passing it over or under the ribbon of the dancer beside them. The circular braid formed around the pole once the dancers finish the first part of the dance is clearly visible and corresponds to the circular braid group.

As promised at the start, I’ve shown you some connections between mathematics and euskal dantzak. The relationship between dance and geometry is probably the most visible. But we’ve also looked at some lesser-known branches of math, such as combinatorics and braid theory, which surprisingly link to some Basque dances. Could it be true that math is really everywhere?