A Visual Tour of Identities for the Padovan Sequence TOM EDGAR AND DAVID NACIN O O ne of the most famous illustrations in mathematics is the image of a spiral traversing points of intersection in a square tiling of the plane, where the square lengths are in correspondence with the Fibonacci sequence. Essentially, only one other similar spiral tiling of regular polygons is possible. On performing the corresponding construction with equilateral triangles instead of squares, as we have done in Figure 1, the side lengths yield the Padovan sequence, given by the recur- rence a n ¼ a n2 þ a n3 : Different sources use various initial conditions; based on the spiral construction, we assume a 0 ¼ 0; a 1 ¼ 1; and a 2 ¼ 1: The first few terms of this sequence are 0; 1; 1; 1; 2; 2; 3; 4; 5; 7; 9; 12; 16; 21; 28; 37; 49; 65; 86: The Padovan sequence can be regarded as Fibonacci’s closest sibling in other ways as well. The characteristic equation for the Fibonacci sequence, x 2 ¼ x þ 1; arises from the desire to split a rectangle into a smaller congruent rectangle and a square, whereas the characteristic equation for the Padovan sequence, x 3 ¼ x þ 1; comes from splitting a square into three congruent rectangles of different sizes. In the way the Fibonacci numbers generate the notorious golden ratio, the Padovan numbers produce what is known as the plastic number. Both sequences represent sums of certain ‘‘shallow diagonals’’ in Pascal’s triangle [1]. Wher- ever Fibonacci numbers appear, Padovan numbers are often right beside them. Ian Stewart [6] apparently deserves credit for naming this collection of numbers after the architect Richard Padovan in 1996. Padovan wrote a 1994 book [3] about the architect Hans van der Laan, the original creator of this sequence, who discovered the numbers in his studies on proportion [2]. A glance at the sequence’s entry (A000931) in the On- Line Encyclopedia of Integer Sequences [5] reveals that interest in the Padovan numbers has grown since Stewart named them, but we have nonetheless found it difficult to find a single resource containing proofs of identities satis- fied by this collection. Because these numbers arise from such a natural construction, one might correctly guess that many identities can be proved purely geometrically. In this paper, we attempt to demonstrate as many Padovan iden- tities as possible in wordless proofs arising directly from rearrangements of the triangles in our original definition. For instance, Figure 2 proves that the Padovan sequence as defined by its spiral construction satisfies the order-3 recurrence stated before as well as an order-5 recurrence. Most of our visual proofs involve areas and lengths of altitudes of equilateral triangles; consequently, we find it best to introduce one constant that we will use throughout. For ease of notation, we set c ¼ ffiffi 3 p =4; making the altitude of an equilateral triangle with side length s equal to 2cs and its area equal to cs 2 : Furthermore, the area of a parallelo- gram with side length s and perpendicular altitude 2ch is given by 2csh. Ó 2021 The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature https://doi.org/10.1007/s00283-021-10076-8