# Volume 9 (2022): Edizione 16 (June 2022)

Dettagli della rivista
Formato
Rivista
eISSN
2182-1976
Pubblicato per la prima volta
16 Apr 2016
Periodo di pubblicazione
2 volte all'anno
Lingue
Inglese

5 Articoli
Accesso libero

#### Flattening the Curve. . . of Spirographs

Pubblicato online: 14 Jun 2022
Pagine: 1 - 20

#### Astratto

The Spirograph is an old and popular toy that produces aesthetically pleasing and fascinating spiral figures. But are spirals all it can make? In playing with a software implementation of the toy, the author chanced upon a variety of shapes that it can make that are different from its well-known repertoire of spirals, in particular, shapes that have a visible flatness and not the curved spiral geometry that we are accustomed to seeing from the Spirograph. This paper reports on these explorations by the author and his delightful discovery of very elegant and simple geometric relationships between the Spirograph’s structural parameters that enable those patterns.

• Spirograph
• polygon
• polygram
• geometry
• gear
• trochoid
• cycloid
Accesso libero

#### Dots-and-Polygons

Pubblicato online: 14 Jun 2022
Pagine: 21 - 42

#### Astratto

Dots-and-Boxes is a popular children’s game whose winning strategies have been studied by Berlekamp, Conway, Guy, and others. In this article we consider two variations, Dots-and-Triangles and Dots-and-Polygons, both of which utilize the same lattice game board structure as Dots-and-Boxes. The nature of these variations along with this lattice structure lends itself to applying Pick’s theorem to calculate claimed area. Several strategies similar to those studied in Dots-and-Boxes are used to analyze these new variations.

Accesso libero

#### Arithmetic Billiards

Pubblicato online: 14 Jun 2022
Pagine: 43 - 54

#### Astratto

Arithmetic billiards show a nice interplay between arithmetics and geometry. The billiard table is a rectangle with integer side lengths. A pointwise ball moves with constant speed along segments making a 45° angle with the sides and bounces on these. In the classical setting, the ball is shooted from a corner and lands in a corner. We allow the ball to start at any point with integer distances from the sides: either the ball lands in a corner or the trajectory is periodic. The length of the path and of certain segments in the path are precisely (up to the factor √2 or 2√2) the least common multiple and the greatest common divisor of the side lengths.

Accesso libero

#### Lagrange was Wrong, Pascal was Right

Pubblicato online: 14 Jun 2022
Pagine: 55 - 64

#### Astratto

In this paper we compare the efficiency of the decimal system to the efficiency of different mixed radix representations. We use as a starting point for our study the duodecimal systems suggested by Pascal and the Maya “Long Count” system. Using the quality index we experimentally show that two slight deviations from the duodecimal system are more efficient than the previous two systems and also than the decimal system.

#### Parole chiave

• numerical systems
• duodecimal system
Accesso libero

#### Musical Modes, their Associated Chords and their Musicality

Pubblicato online: 14 Jun 2022
Pagine: 65 - 70

#### Astratto

In this paper we present a mathematical way of defining musical modes and we define the musicality of a mode as a product of three diferent factors. We conclude by classyfing the modes which are most musical according to our definition.