Spiral Of Theodorus
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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the spiral of Theodorus (also called ''square root spiral'', ''Einstein spiral'', ''Pythagorean spiral'', or ''Pythagoras's snail'') is a
spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s, placed edge-to-edge. It was named after
Theodorus of Cyrene Theodorus of Cyrene ( el, Θεόδωρος ὁ Κυρηναῖος) was an ancient Greek mathematician who lived during the 5th century BC. The only first-hand accounts of him that survive are in three of Plato's dialogues: the ''Theaetetus'', th ...
.


Construction

The spiral is started with an
isosceles In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
right triangle, with each
leg A leg is a weight-bearing and locomotive anatomical structure, usually having a columnar shape. During locomotion, legs function as "extensible struts". The combination of movements at all joints can be modeled as a single, linear element ca ...
having unit
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
. Another right triangle is formed, an automedian right triangle with one leg being the
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
of the prior triangle (with length the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
) and the other leg having length of 1; the length of the hypotenuse of this second triangle is the
square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as \sqrt or 3^. It is more precisely called the principal square root of 3 to distinguish it from the negative nu ...
. The process then repeats; the nth triangle in the sequence is a right triangle with the side lengths \sqrt and 1, and with hypotenuse \sqrt. For example, the 16th triangle has sides measuring 4=\sqrt, 1 and hypotenuse of \sqrt.


History and uses

Although all of Theodorus' work has been lost,
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
put Theodorus into his dialogue '' Theaetetus'', which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
by means of the Spiral of Theodorus. Plato does not attribute the irrationality of the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.


Hypotenuse

Each of the triangles' hypotenuses h_n gives the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of the corresponding
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
, with h_1=\sqrt. Plato, tutored by Theodorus, questioned why Theodorus stopped at \sqrt. The reason is commonly believed to be that the \sqrt hypotenuse belongs to the last triangle that does not overlap the figure.


Overlapping

In 1958, Kaleb Williams proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a line, they will never pass through any of the other vertices of the total figure.


Extension

Theodorus stopped his spiral at the triangle with a hypotenuse of \sqrt. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.


Growth rate


Angle

If \varphi_n is the angle of the nth triangle (or spiral segment), then: \tan\left(\varphi_n\right)=\frac. Therefore, the growth of the angle \varphi_n of the next triangle n is: \varphi_n=\arctan\left(\frac\right). The sum of the angles of the first k triangles is called the total angle \varphi(k) for the kth triangle. It grows proportionally to the square root of k, with a bounded correction term c_2: \varphi\left (k\right)=\sum_^k\varphi_n = 2\sqrt+c_2(k) where \lim_ c_2(k)= - 2.157782996659\ldots ().


Radius

The growth of the radius of the spiral at a certain triangle n is \Delta r=\sqrt-\sqrt.


Archimedean spiral

The Spiral of Theodorus
approximate An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
s the
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
. Just as the distance between two windings of the Archimedean spiral equals
mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
\pi, as the number of spins of the spiral of Theodorus approaches
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
, the distance between two consecutive windings quickly approaches \pi. The following is a table showing of two windings of the spiral approaching pi: As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to \pi.


Continuous curve

The question of how to
interpolate In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a n ...
the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered in by analogy with Euler's formula for the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
as an
interpolant In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a ...
for the
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
function.
Davis Davis may refer to: Places Antarctica * Mount Davis (Antarctica) * Davis Island (Palmer Archipelago) * Davis Valley, Queen Elizabeth Land Canada * Davis, Saskatchewan, an unincorporated community * Davis Strait, between Nunavut and Gre ...
found the function T(x) = \prod_^\infty \frac \qquad ( -1 < x < \infty ) which was further studied by his student
Leader Leadership, both as a research area and as a practical skill, encompasses the ability of an individual, group or organization to "lead", influence or guide other individuals, teams, or entire organizations. The word "leadership" often gets vi ...
and by Iserles (in an appendix to ). An axiomatic characterization of this function is given in as the unique function that satisfies the
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
f(x+1) = \left( 1 + \frac\right) \cdot f(x), the initial condition f(0) = 1, and
monotonicity In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
in both
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
and modulus; alternative conditions and weakenings are also studied therein. An alternative derivation is given in . An analytic continuation of Davis' continuous form of the Spiral of Theodorus which extends in the opposite direction from the origin is given in . In the figure the nodes of the original (discrete) Theodorus spiral are shown as small green circles. The blue ones are those, added in the opposite direction of the spiral. Only nodes n with the integer value of the polar radius r_n=\pm\sqrt are numbered in the figure. The dashed circle in the coordinate origin O is the circle of curvature at O.


See also

*
Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance f ...
*
List of spirals This list of spirals includes named spirals that have been described mathematically. See also * Catherine wheel (firework) * List of spiral galaxies * Parker spiral * Spirangle * Spirograph Spirograph is a geometric drawing device that ...


References


Further reading

* * * * {{Spirals Theodorus Pythagorean theorem Pi