Gnomon (figure)#Isosceles triangles

Figurate numbers were a concern of Pythagorean mathematics, and Pythagoras 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.{{citation|title=Figurate Numbers|first1=Elena|last1=Deza|author1-link=Elena Deza|first2=Michel|last2=Deza|author2-link=Michel Deza|publisher=World Scientific|year=2012|isbn=9789814355483|page=3|url=https://books.google.com/books?id=cDxYdstLPz4C&pg=PA3}}.

For example, the gnomon of the square number is the odd number, of the general form 2n + 1, n = 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this:

$~~~~~~~~\backslash begin\{matrix\}$

1&2&3&4&5&6&7&8\\

2&2&3&4&5&6&7&8\\

3&3&3&4&5&6&7&8\\

4&4&4&4&5&6&7&8\\

5&5&5&5&5&6&7&8\\

6&6&6&6&6&6&7&8\\

7&7&7&7&7&7&7&8\\

8&8&8&8&8&8&8&8

\end{matrix}

To transform from the n-square (the square of size n) to the (n + 1)-square, one adjoins 2n + 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 n²; the figure illustrates {{nowrap|1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 {{=}} 64 {{=}} 8².}}

File:Nicomachus_theorem_3D.svg

Applying the same technique to a multiplication table gives the Nicomachus theorem, proving that each squared triangular number is a sum of cubes.{{citation|first=T. Sundara|last=Row|title= Geometric Exercises in Paper Folding|title-link= Geometric Exercises in Paper Folding|year=1893|location=Madras|publisher=Addison|at=[https://archive.org/details/geometricexerci00raogoog/page/n61/mode/2up pp. 46–48]}}.

In an acute isosceles triangle, 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 which case the acute isosceles triangle is the golden triangle and its gnomon is the golden gnomon.{{citation|contribution=The Golden Triangle|first=Arthur L.|last=Loeb|title=Concepts & Images: Visual Mathematics|publisher=Springer|series=Design Science Collection|year=1993|pages=179–192|doi=10.1007/978-1-4612-0343-8_20|isbn=978-1-4612-6716-4}}

Conversely, the acute golden triangle can be the gnomon of the obtuse golden triangle in an exceptional reciprocal exchange of roles{{cite web|url=http://www.maecla.it/tartapelago/museo/gnomoni/indexen.htm |title=Gnomons collection|first=Giorgio|last=Pietrocola |date=2005 |website=Maecla }}

File:Golden triangle (math).svg|golden triangle 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

File:Gnomons from Tartapelago 08.gif|The acute golden triangle is gnomon of a octagon

File:Gnomons from Tartapelago 07.gif|The obtuse golden triangle is the gnomon of a enneagon

A metaphor based around the geometry of a gnomon plays an important role in the literary analysis of James Joyce's Dubliners, 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.{{citation|title=The Gnomonic Clue to James Joyce's Dubliners|first=Gerhard|last=Friedrich|journal=Modern Language Notes|volume=72|issue=6|year=1957|pages=421–424|doi=10.2307/3043368 |jstor=3043368}}.{{citation|title=Gnomon Is an Island: Euclid and Bruno in Joyce's Narrative Practice|first=David|last=Weir|journal=James Joyce Quarterly|volume=28|issue=2|year=1991|pages=343–360|jstor=25485150}}.{{citation|title=The Perspective of Joyce's Dubliners|journal=College English|first=Gerhard|last=Friedrich|volume=26|issue=6|year=1965|pages=421–426|doi=10.2307/373448 |jstor=373448}}.{{citation|contribution=Fragment and totality|first=Klaus|last=Reichert|contribution-url=https://books.google.com/books?id=_hr--JnJDlAC&pg=PA86|pages=86–87|title=New Alliances in Joyce Studies: When It's Aped to Foul a Delfian|editor-first=Bonnie Kime|editor-last=Scott|publisher=University of Delaware Press|year=1988|isbn=9780874133288}}

Gnomon shapes are also prominent in Arithmetic Composition I, an abstract painting by Theo van Doesburg.{{citation|contribution=From Art to Mathematics in the Paintings of Theo van Doesburg|first1=Paola|last1=Vighi|first2=Igino|last2=Aschieri|title=Applications of Mathematics in Models, Artificial Neural Networks and Arts|publisher=Springer|year=2010|pages=601–610|doi=10.1007/978-90-481-8581-8_27|series=Mathematics and Society|editor1-first=Vittorio|editor1-last=Capecchi|editor2-first=Massimo|editor2-last=Buscema|editor3-first=Pierluigi|editor3-last=Contucci|editor4-first=Bruno|display-editors = 3 |editor4-last=D'Amore|isbn=978-90-481-8580-1}}.

There is also a very short geometric fairy tale illustrated by animations where gnomons play the role of invaders.{{cite web|url=http://www.pietrocola.eu/maecla/tartapelago/fiabe/Goldenking/index.htm |title=Golden King and the invasion of the gnomons|first=Giorgio|last=Pietrocola |date=2005 |website=Maecla }}.

• Theorem of the gnomon

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Category:Figurate numbers