On the meaning of "gnomon":
change invariance
July 2020     by Giorgio Pietrocola

quote from the book
Paolo Zellini, Gnomon A Survey on the Number, Adelphi, 1999
The term "gnomon" had several meanings in Greece, one of which was certainly astronomical. Following a technique of probable Babylonian origin (Herodotus, Stories, II, 109) time was measured by the shadow cast by a stylus, the gnomon, planted perpendicularly on a horizontal surface. The stylus and its shadow formed a square shape (from which the expression "conforming to the gnomon" Enopide of Chio called the perpendicular a straight line from an external point) analogous to the frame of growth of the square numbers.
The same term "gnomon" denoted any linear or flat size that could be added or taken away from certain geometric figures while keeping their shape unchanged. The gnomon could thus apply to the theory of similarity
...
Heron defined a gnomon, in general, as what, "added to any entity, number or figure, makes it all similar to the entity to which it was added". (Heronis Definitiones cum variis collectionibus, Def. 58, and J.L. Heiberg, vol.IV)
The gnomon therefore in its geometric meaning taken from tradition, is not a particular entity but rather a relationship between two entities.
The relationship
"A is the gnomon of B"
it is true when A, suitably added or subtracted from B, gives a result similar to B (i.e. leaves its shape unchanged).

Here are some examples illustrated by special animations:

1. A square with a side equal to the longer side of a golden rectangle is its gnomon
 


2. Another particular square, protagonist of the next animation, is the gnomon of the accumulation figure of infinite squares contained in a rectangle
 

3. Two "pacmans" are respectively gnomons of a golden hexagon and an octagon.

4. Two particular triangles are the gnomons of a snail and of the nautilus respectively

5. A trapezium and a golden triangle are gnomons of the respective accumulation figures

6. The relation "to be the gnomon of" is not commutative however in a very particular case, the only one that I know of, this commutativity exists in fact the two golden triangles are mutually each other's gnomon.

7. Increasingly voracious fish are gnomons of their curvilinear figure of infinite accumulation

8. Other examples by pressing the buttons on the left.