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To transform from the ''n-square'' (the square of size ''n'') to the (''n'' + 1)-square, one adjoins 2''n'' + 1 elements: one to the end of each row (''n'' elements), one to the end of each column (''n'' elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above figure. This gnomonic technique also provides a proof that the sum of the first ''n'' odd numbers is ''n''2; the figure illustrates Applying the same technique to a
File:Golden triangle (math).svg,
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 ...
, a gnomon is a plane figure formed by removing a similar parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
from a corner of a larger parallelogram; or, more generally, a figure that, added to a given figure, makes a larger figure of the same shape.
Building figurate numbers
Figurate numbers
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean
* polygon ...
were a concern of Pythagorean mathematics, and Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
is credited with the notion that these numbers are generated from a ''gnomon'' or basic unit. The gnomon is the piece which needs to be added to a figurate number to transform it to the next bigger one.
For example, the gnomon of the square number is the odd number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
41 ...
, of the general form 2''n'' + 1, ''n'' = 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this:To transform from the ''n-square'' (the square of size ''n'') to the (''n'' + 1)-square, one adjoins 2''n'' + 1 elements: one to the end of each row (''n'' elements), one to the end of each column (''n'' elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above figure. This gnomonic technique also provides a proof that the sum of the first ''n'' odd numbers is ''n''2; the figure illustrates Applying the same technique to a
multiplication table
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.
The decimal multiplication table was traditionally taught as an essenti ...
proves that each squared triangular number
In number theory, the sum of the first cubes is the square of the th triangular number. That is,
:1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2.
The same equation may be written more compactly using the mathematical notation for summa ...
is a sum of cubes.
Isosceles triangles
In anacute
Acute may refer to:
Science and technology
* Acute angle
** Acute triangle
** Acute, a leaf shape in the glossary of leaf morphology
* Acute (medicine), a disease that it is of short duration and of recent onset.
** Acute toxicity, the adverse eff ...
isosceles triangle
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 ...
, it is possible to draw a similar but smaller triangle, one of whose sides is the base of the original triangle. The gnomon of these two similar triangles is the triangle remaining when the smaller of the two similar isosceles triangles is removed from the larger one. The gnomon is itself isosceles if and only if the ratio of the sides to the base of the original isosceles triangle, and the ratio of the base to the sides of the gnomon, is the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
, in which case the acute isosceles triangle is the golden triangle
Golden Triangle may refer to:
Places
Asia
* Golden Triangle (Southeast Asia), named for its opium production
* Golden Triangle (Yangtze), China, named for its rapid economic development
* Golden Triangle (India), comprising the popular tourist ...
and its gnomon is the golden gnomon
A golden triangle, also called a sublime triangle,
is an isosceles triangle in which the duplicated side is in the golden ratio \varphi to the base side:
: = \varphi = \approx 1.618~034~.
Angles
* The vertex angle is:
::\theta = 2\arcsin = 2\ar ...
.
Conversely, the acute golden triangle can be the gnomon of the obtuse golden triangle in an exceptional reciprocal exchange of roles
golden triangle
Golden Triangle may refer to:
Places
Asia
* Golden Triangle (Southeast Asia), named for its opium production
* Golden Triangle (Yangtze), China, named for its rapid economic development
* Golden Triangle (India), comprising the popular tourist ...
partitioned into a smaller golden triangle and the (obtuse) golden gnomon
File:Goldentrianges_3s108_Tartapelago.gif, The obtuse golden triangle is the gnomon of acute golden triangle
File:Gnomons_from_Tartapelago_06.gif, The acute golden triangle is the gnomon of the obtuse golden triangle
Metaphor and symbolism
A metaphor based around the geometry of a gnomon plays an important role in the literary analysis ofJames Joyce
James Augustine Aloysius Joyce (2 February 1882 – 13 January 1941) was an Irish novelist, poet, and literary critic. He contributed to the modernist avant-garde movement and is regarded as one of the most influential and important writers of ...
's ''Dubliners
''Dubliners'' is a collection of fifteen short stories by James Joyce, first published in 1914. It presents a naturalistic depiction of Irish middle class life in and around Dublin in the early years of the 20th century.
The stories were writt ...
'', involving both a play on words between "paralysis" and "parallelogram", and the geometric meaning of a gnomon as something fragmentary, diminished from its completed shape.
Gnomon shapes are also prominent in ''Arithmetic Composition I'', an abstract painting by Theo van Doesburg
Theo van Doesburg (, 30 August 1883 – 7 March 1931) was a Dutch artist, who practiced painting, writing, poetry and architecture. He is best known as the founder and leader of De Stijl. He was married to artist, pianist and choreographer Nelly ...
.
There is also a very short geometric fairy tale illustrated by animations where gnomons play the role of invaders.
See also
* Theorem of the gnomonReferences
{{DEFAULTSORT:Gnomon (Figure) Figurate numbers