In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 the Padovan cuboid spiral is the spiral

Spiral

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

 created by joining the diagonals of faces of successive cuboid

Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

s added to a unit cube. The cuboids are added sequentially so that the resulting cuboid has dimensions that are successive Padovan numbers

Padovan sequence

The Padovan sequence is the sequence of integers P defined by the initial valuesP=P=P=1,and the recurrence relationP=P+P.The first few values of P are...

.

The first cuboid is 1x1x1. The second is formed by adding to this a 1x1x1 cuboid to form a 1x1x2 cuboid. To this is added a 1x1x2 cuboid to form a 1x2x2 cuboid.
This pattern continues, forming in succession a 2x2x3 cuboid, a 2x3x4 cuboid etc. Joining the diagonals of the exposed end of each new added cuboid creates a spiral

Spiral

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

 (seen as the black line in the figure). The points on this spiral

Spiral

In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...

 all lie in the same plane.

The cuboids are added in a sequence that adds to the face in the positive y direction, then the positive x direction, then the positive z direction. This is followed by cuboids added in the negative y, negative x and negative z directions. Each new cuboid added has a length and width that matches the length and width of the face being added to. The height of the nth added cuboid is the nth Padovan number.

Connecting alternate points where the spiral bends creates a series of triangles, where each triangle has two sides that are successive Padovan numbers and that has an acute angle of 120 degrees between these two sides.

As the spiral unfolds, the triangles approach a form that has angles of 120 degrees, 34.61 degrees and 25.39 degrees.