Hans Walser, [20250529]

Golden Section

Idea and suggestion: Shingo Nakanishi, Japan

1 What it's about

Different approaches to a sequence of numbers in which the golden ratio appears as the first non-trivial number.

2 Equal decimal places

Which positive real numbers in the decimal fraction expansion have the same digits after the decimal point as their reciprocal?

The trivial example is the number 1.

Two positive numbers have the same digits after the decimal point if and only if their difference is a natural number. For our problem, this results in the equation:

x – 1/x = n, n ∈ ℕ

This leads to the quadratic equation:

x² – nx – 1 = 0

The positive solution is:

x = (n + √(n² + 4))/2

For the reciprocal we get:

1/x = (–n + √(n² + 4))/2

Table 1 gives the first solutions. The solutions are given exactly and in decimal approximation.

n	x	x	1/x	1/x
0	1	1.	1	1.	Trivial solution
1	1/2 + (1/2)*5^(1/2)	1.618033988	– 1/2 + (1/2)*5^(1/2)	.6180339880	Golden section
2	1 + 2^(1/2)	2.414213562	– 1 + 2^(1/2)	.414213562
3	3/2 + (1/2)*13^(1/2)	3.302775638	– 3/2 + (1/2)*13^(1/2)	.302775638
4	2 + 5^(1/2)	4.236067977	– 2 + 5^(1/2)	.236067977
5	5/2 + (1/2)*29^(1/2)	5.192582404	– 5/2 + (1/2)*29^(1/2)	.192582404

Tab. 1: First solutions

3 Cut off squares

We are looking for rectangles of height 1 and length x such that, after cutting off n unit squares, a rectangle similar to the original rectangle remains, but standing upright.

Figure 1 shows the situation for n = 3.

Fig. 1: Cut off 3 squares

It is x = n + 1/x. The similarity condition yields:

(n + 1/x)/1 = 1/(1/x)

Hence:

n + 1/x = x

This in turn results in the quadratic equation:

x² – nx – 1 = 0

Figure 2 shows the first examples.

Fig. 2: Rectangles

The trivial example is the square where no square is cut off and the remaining rectangle is the square standing on edge. — These trivial examples are often difficult to understand. Therefore, the simplest non-trivial example should always be discussed first.

Figure 3 illustrates the similarity by subdividing the remaining rectangle into n smaller squares and a second-degree remaining rectangle.

Fig. 3: Subdivision of the remaining rectangle

The remaining rectangle of second degree can also be subdivided (Fig. 4).

Fig. 4: Next turn

And so it goes on (Fig. 5 and Fig. 6).

Fig. 5: Next turn

Fig. 6: Next turn

4 Spirals

Instead of iterating the figures over the number n of truncated squares, as in the section above, we can also keep n fixed and iterate over the number k of continued subdivisions. Figure 7 shows the situation for the case n = 1. This creates a spiral arrangement of ever-shrinking squares. This is sometimes referred to as the golden spiral. Theoretically, the number of subdivisions is infinite. However, after about 8 subdivision steps, no further change is visually detectable.

Fig. 7: Golden spiral

Figure 8 shows the situation for n = 2. The result is a spiral composed of pairs of equal squares. It is sometimes referred to as a silver spiral.

Fig. 8: Spiral composed of pairs of squares

Figure 9 shows the situation for n = 3. The result is a spiral composed of triples of equal squares.

Fig. 9: Spiral composed of triples of squares

The situation is exciting for n = 0. Nothing happens there (Fig. 10). It is a spiral composed of zerotuples of equal squares.

Fig. 10: Spiral composed of zerotuples of squares

5 In the square grid

The desired rectangles can also be constructed using a compass and a grid of squares. We illustrate the procedure for n = 3.

We start with a 2×n grid of squares (Fig. 11). Additionally, we draw the midline and the circumcircle.

Fig. 11: Square grid

The intersection points of the circumcircle with the midline and two diametrical corner points of the grid define the desired rectangle (Fig. 12).

Fig. 12: Rectangle

Figure 13 illustrates the truncation of the rectangle into n squares. The truncated yellow squares are larger than the squares of the original grid.

Fig. 13: Cut off squares

Animation 14 shows the situation for n from 0 to 5.

Fig. 14: Animation

6 Square grid and parabola

Instead of the circumcircle, we can also work with a standard parabola. The vertex curvature of this parabola should be equal to the incircles of the grid squares (Fig. 15).

Fig. 15: Square grid and parabola

Figure 16 shows an animation.

Fig. 16: Animation

Weblinks

Shingo Nakanishi

https://www.oit.ac.jp/labs/center/nakanishi/img2/258.SVG

Hans Walser: Goldener Schnitt

https://walser-h-m.ch/hans/Miniaturen/G/Goldener_Schnitt_30/Goldener_Schnitt_30.html

References

Walser, Hans (2024): Spirals, Helical Lines, and Spiral-Like Figures. Mathematical Playfulness in Two and Three Dimensions. Springer.
ISBN 978-3-662-68930-1, ISBN 978-3-662-68931-8 (eBook)
https://doi.org/10.1007/978-3-662-68931-8

Walser, Hans (2024): The Golden Ratio. Geometric and Number Theoretical Considerations. Springer. ISBN 978-3-662-69889-1, ISBN 978-3-662-69890-7 (eBook)
https://doi.org/10.1007/978-3-662-69890-7