Spiral Tiling from Integer Sequences


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Sequence is a rundown of a set of numbers in a particular arrangement. It contains values, similar to that of a set. Be that as it may, a sequence can also contain the same values at some dissimilar areas. Hence, a pattern is an important component of sequences.

Many sequences are already existing, the likes of Lucas, Fibonacci, arithmetic, geometric, square. There are some sequences that have a known formula in finding the nth term such as: arithmetic, geometric, harmonic and Fibonacci. An integer sequence is a sequence of integers that may be stated obviously by giving a formula for its nth term, or indirectly by giving a relationship between its terms. For example, the sequence 1, 1, 2, 3, 5, 8, 13,… is formed by starting with 1 and 1 then adding any two consecutive terms to get the next one, an indirect description, While the sequence 0, 2, 6, 12, 20, 30,… is formed according to the formula n2 – n for the nth term: an obvious description.

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Tiling includes a segment of room into pieces that are consistent to one another; however there has been some investigation of similitude tiling in which this coinciding necessity is loose, with the goal that the tiles require just be similar to one another. One method of arranging these tiles is in spiral. A popular example of a spiral tiling is the figure below. This shape is frequently called the “golden rectangle”. By letting Sn be the length of the side of the nth square in this spiral.

By examining the length of the edges, we see that it satisfy the Fibonacci sequence Sn = Sn-1 + Sn-2. Similarly, by letting the Greek letter phi (φ) be the ratio Sn/Sn-1, it follows that φ is a root of the equation x2- x-1=0. The roots of this equation is known as the “golden ratio”. (1 ± √5)÷2 = 1. 618… or -0. 618…Another lesser known spiral tiling is the analogous construction for equilateral triangles. It is a sequence of equilateral triangles and each triangles’ height follow a formula of hk = hk-1 + hk-5 or hk = hk-2 + hk-3. It follows that the sequence has two equations x5 – x4 – 1= 0 and x3 – x – 1 = 0. It can be attributed that the sequence has two equations because the expression: x5 – x4 – 1 is equal to (x3 – x – 1)(x2 – x + 1). As shown above, a plane can be tiled by a spiral of geometric figures that follow a certain integer sequence. Hence, there is a need for study on these fields more. This study will try to connect the different concepts of number theory with that of tiling theory.

Statement of the Problem

In this study, we sought to know to provide a systematic construction of spiral tilings from a given integer sequence.

In this work, The Researchers aim to fulfil the following specific objectives:

a. To know which among the known integer sequences lead to spiral tilingsb.

To device an algorithm on how to construct spiral tilings from integer sequences.

c. To determine which polygons can be used in creating spiral tilings from integer sequences.

Significance of the Study

This study will try to make a connection between the different concepts of number theory and tiling theory. Hence, it will contribute to the body of knowledge in mathematics, specifically in the field of spiral tiling and integer sequences.

Scope and Limitations

This study focuses on the tilings of the Euclidean plane. The tilings will be constructed from the following integer sequences: Fibonacci sequence, Lucas sequence, and other variants of the Fibonacci sequences.

Conceptual Framework

The Researchers will determine which integer sequences lead to spiral tilings.


A kth term in a Fibonacci sequence with a value of nk corresponds to a square of dimension nk x nk. Hence, it is possible to construct a spiral tiling by using squares. The spiral tiling formed by the Fibonacci sequence is the golden rectangle.

Definition of Terms

On this part of the study, significant terminologies that are being used throughout the study are listed and defined operationally for the purpose of reader’s unambiguity.

Fibonacci sequence is an integer sequence that has a corresponding spiral tiling that is formed with squares; the corresponding spiral tiling formed by the Fibonacci sequence is the golden rectangle. Fibonacci sequence can also refer to an integer sequence, such that the next term can be found by adding the two consecutive terms.

Golden ratio is the roots of the characteristic equation of the Fibonacci sequence.

Golden rectangle is composed of squares that have a side length that corresponds to the terms of the Fibonacci sequence. It is also the corresponding spiral tiling of the Fibonacci sequence. Integer sequences are the factors that are being determined whether they will lead to spiral tilings. It can also refer to any sequence of integers that was stated by giving a formula for its nth term or by giving the relationship of its terms.

Lucas sequence is another integer sequence that is being determined if it has a corresponding spiral tiling. Other variants of the Fibonacci sequence refers to other integer sequences that are being determined if it has corresponding spiral tiling and are integer sequences whose terms are defined by to be the sum of its two immediate previous terms. Spiral Tilings are tilings that are arranged in a spiral manner and that corresponds to a certain integer sequence.

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