Gnomon (figure)

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A gnomon Gnomon.svg
A gnomon

In geometry, a gnomon is a plane figure formed by removing a similar parallelogram 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. [1]

Contents

Building figurate numbers

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. [2]

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:



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 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 82.

First five terms of Nichomachus's theorem Nicomachus theorem 3D.svg
First five terms of Nichomachus's theorem

Applying the same technique to a multiplication table gives the Nicomachus theorem, proving that each squared triangular number is a sum of cubes. [3]

Isosceles triangles

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. [4] Conversely, the acute golden triangle can be the gnomon of the obtuse golden triangle in an exceptional reciprocal exchange of roles [5]

Metaphor and symbolism

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. [6] [7] [8] [9]

Gnomon shapes are also prominent in Arithmetic Composition I, an abstract painting by Theo van Doesburg. [10]

There is also a very short geometric fairy tale illustrated by animations where gnomons play the role of invaders [11]

See also

Related Research Articles

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<span class="mw-page-title-main">Triangle</span> Shape with three sides

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<span class="mw-page-title-main">Gnomon</span> Part of a sundial that casts a shadow

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<span class="mw-page-title-main">Shape</span> Form of an object or its external boundary

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<span class="mw-page-title-main">Pentagram</span> Five-pointed star polygon

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In geometry, a trapezoid in North American English, or trapezium in British English, is a quadrilateral that has at least one pair of parallel sides.

<span class="mw-page-title-main">Thales's theorem</span> Angle formed by a point on a circle and the 2 ends of a diameter is a right angle

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

<span class="mw-page-title-main">Isosceles triangle</span> Triangle with at least two sides congruent

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<span class="mw-page-title-main">Centered hexagonal number</span> Number that represents a hexagon with a dot in the center

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<span class="mw-page-title-main">Special right triangle</span> Right triangle with a feature making calculations on the triangle easier

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<span class="mw-page-title-main">Golden triangle (mathematics)</span> Isosceles triangle in which the duplicated side is in the golden ratio to the base side

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<span class="mw-page-title-main">Disphenoid</span> Tetrahedron whose faces are all congruent

In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are isotetrahedron, sphenoid, bisphenoid, isosceles tetrahedron, equifacial tetrahedron, almost regular tetrahedron, and tetramonohedron.

<span class="mw-page-title-main">Pythagorean theorem</span> Relation between sides of a right triangle

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

<span class="mw-page-title-main">Rep-tile</span> Shape subdivided into copies of itself

In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

<span class="mw-page-title-main">Acute and obtuse triangles</span> Triangles without a right angle

An acute triangle is a triangle with three acute angles. An obtuse triangle is a triangle with one obtuse angle and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles — triangles that are not right triangles because they do not have a 90° angle.

<span class="mw-page-title-main">Theorem of the gnomon</span> Certain parallelograms occurring in a gnomon have areas of equal size

The theorem of the gnomon states that certain parallelograms occurring in a gnomon have areas of equal size.

References

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  2. Deza, Elena; Deza, Michel (2012), Figurate Numbers, World Scientific, p. 3, ISBN   9789814355483 .
  3. Row, T. Sundara (1893), Geometric Exercises in Paper Folding, Madras: Addison, pp. 46–48 .
  4. Loeb, Arthur L. (1993), "The Golden Triangle", Concepts & Images: Visual Mathematics, Design Science Collection, Springer, pp. 179–192, doi:10.1007/978-1-4612-0343-8_20, ISBN   978-1-4612-6716-4
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  8. Friedrich, Gerhard (1965), "The Perspective of Joyce's Dubliners", College English, 26 (6): 421–426, doi:10.2307/373448, JSTOR   373448 .
  9. Reichert, Klaus (1988), "Fragment and totality", in Scott, Bonnie Kime (ed.), New Alliances in Joyce Studies: When It's Aped to Foul a Delfian, University of Delaware Press, pp. 86–87, ISBN   9780874133288
  10. Vighi, Paola; Aschieri, Igino (2010), "From Art to Mathematics in the Paintings of Theo van Doesburg", in Capecchi, Vittorio; Buscema, Massimo; Contucci, Pierluigi; et al. (eds.), Applications of Mathematics in Models, Artificial Neural Networks and Arts, Mathematics and Society, Springer, pp. 601–610, doi:10.1007/978-90-481-8581-8_27, ISBN   978-90-481-8580-1 .
  11. Pietrocola, Giorgio (2005). "Golden King and the invasion of the gnomons". Maecla..