Gnomon_(figure)

Gnomon (figure)

Gnomon (figure)

Figure that, added to a given figure, makes a larger figure of the same shape

For the part of a sundial and history, see Gnomon. For other uses, see Gnomon (disambiguation).

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]

A gnomon

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

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]

  • golden triangle partitioned into a smaller golden triangle and the (obtuse) golden gnomon
  • The obtuse golden triangle is the gnomon of acute golden triangle
  • The acute golden triangle is the gnomon of the obtuse golden triangle
  • The acute golden triangle is gnomon of a octagon
  • The obtuse golden triangle is the gnomon of a enneagon

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

References

  1. [1]
    Gazalé, Midhat J. (1999), "Gnomon: From Pharaohs to Fractals", European Journal of Physics, 20 (6), Princeton University Press: 523, Bibcode:1999EJPh...20..523G, ISBN 9780691005140.
  2. [2]
    Deza, Elena; Deza, Michel (2012), Figurate Numbers, World Scientific, p. 3, ISBN 9789814355483.
  3. [3]
    Row, T. Sundara (1893), Geometric Exercises in Paper Folding, Madras: Addison, pp. 46–48.
  4. [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
  5. [5]
    Pietrocola, Giorgio (2005). "Gnomons collection". Maecla.
  6. [6]
    Friedrich, Gerhard (1957), "The Gnomonic Clue to James Joyce's Dubliners", Modern Language Notes, 72 (6): 421–424, doi:10.2307/3043368, JSTOR 3043368.
  7. [7]
    Weir, David (1991), "Gnomon Is an Island: Euclid and Bruno in Joyce's Narrative Practice", James Joyce Quarterly, 28 (2): 343–360, JSTOR 25485150.
  8. [8]
    Friedrich, Gerhard (1965), "The Perspective of Joyce's Dubliners", College English, 26 (6): 421–426, doi:10.2307/373448, JSTOR 373448.
  9. [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. [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. [11]
    Pietrocola, Giorgio (2005). "Golden King and the invasion of the gnomons". Maecla..
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