Triangle

The equilateral triangle is the first construction in Euclid, and it takes two circles. Given a segment AB, strike a circle at A through B and a circle at B through A, and call one of their crossings C. Then AC and BC are both radii, so both equal AB, and ABC is equilateral. That is the whole of Proposition I.1.

The triangle is constructible in the technical sense that matters. The Gauss-Wantzel theorem states that a regular n-gon can be drawn with compass and straightedge exactly when n is a power of 2 multiplied by a product of distinct Fermat primes. Fermat primes are primes of the form 2 to the power 2 to the power k, plus one; the only ones known are 3, 5, 17, 257 and 65537. Three is the first of them, so the equilateral triangle qualifies, and so does every doubling of it: hexagon, 12-gon, 24-gon and onward, obtained by bisecting arcs.

Other triangles are equally accessible. Euclid's Book I constructs a triangle from three given sides at Proposition 22, provided any two of them together exceed the third. Bisecting an angle, dropping a perpendicular and copying an angle are all triangle work.

Three points not lying in a straight line determine exactly one triangle, exactly one circle through them, and exactly one plane containing them. This is why the triangle is the natural unit for describing a surface, and why computer graphics decomposes everything into triangles: three vertices are always coplanar, so a triangle is guaranteed flat in a way that a quadrilateral is not.

The angles of a plane triangle sum to two right angles. That is Euclid I.32, and it depends on the parallel postulate. On a sphere the sum exceeds 180 degrees; on a hyperbolic surface it falls short. The triangle is thus the figure that most directly registers what kind of space it has been drawn in.

Fix the three side lengths of a triangle and the shape is fixed with them. No flexing is possible, because three sides determine the three angles completely, which is the content of the side-side-side congruence result at Euclid I.8. No other polygon has this property. A quadrilateral with four fixed side lengths can be pushed into a parallelogram and back again; a hexagon collapses more readily still.

The consequence is structural rather than symbolic. Bridge trusses, roof trusses, pylons, bicycle frames, scaffolding and space frames are all triangulated, and the diagonal brace added to a rectangular frame does nothing except convert one flexible quadrilateral into two rigid triangles. Buckminster Fuller's geodesic domes are triangulated spheres for the same reason. The engineering term is that a triangulated frame is statically determinate: the forces in it can be resolved by geometry alone, member by member.

Triangles also tile the plane, and unusually, any triangle tiles it, not merely the equilateral one. Take any triangle at all, rotate a copy through 180 degrees, and the two together form a parallelogram, which tiles trivially. The same holds for any quadrilateral. It stops being true at five sides. Among regular polygons only the triangle, square and hexagon tile alone, because the interior angle must divide 360 degrees exactly, and 60, 90 and 120 are the only regular interior angles that do.

Triangular numbers, 1, 3, 6, 10, 15 and so on, count the dots in a triangular array. The fourth of them, ten arranged in four rows, is the Pythagorean tetractys, which the ancient sources describe as an object of oath and reverence within the school.

Threefold structures occur across enough unrelated traditions that the interesting question is not whether they exist but how differently they work.

In Christianity the triangle is the standard visual shorthand for the Trinity, most explicitly in the medieval diagram called the Scutum Fidei or shield of faith, which sets Father, Son and Spirit at three corners with God at the centre and labels the connecting bars to state precisely which identities hold and which do not. It is a piece of doctrinal engineering as much as an image, designed to head off the errors the councils had spent centuries naming. The eye set within a triangle, common on eighteenth-century church furnishings and later on Masonic and civic emblems, is a related but much looser use.

In Hindu tantra the upward triangle is read as Shiva and the downward as Shakti, and the Sri Yantra is built from nine interlocking triangles, four upward and five downward, converging on the central bindu. The reading is directional and gendered in a way the Christian use is not.

Egyptian religion grouped gods in threes, Amun, Mut and Khonsu at Thebes, or Osiris, Isis and Horus, and Egyptologists generally read these as family or local groupings rather than as any statement about unity within plurality. Reading them as a proto-Trinity is a nineteenth-century habit that the evidence does not support.

Alchemical and Western esoteric traditions used the triangle as the elemental sign: point up for fire, point down for water, each with a horizontal bar added for air and earth respectively. That scheme is medieval and early modern rather than ancient, though it is frequently presented as far older than it is.