## Abstract

The golden ratio and the plastic number are both so-called morphic numbers that have been studied in the past in various scientific domains, in particular in architecture. Based on the golden ratio, the concept of a golden angle has been defined for a circle in two-dimensional Euclidean space. However, at present, there exists no three-dimensional analog based on the plastic number. In this paper, the concept of morphic angles will be introduced, based on morphic numbers. New definitions will be proposed for these angles that are applicable for nondegenerated quadratic curves and surfaces of revolution, respectively.

## Keywords

Golden ratio Plastic number Golden morphic angle Plastic angle Plastic morphic angle## Introduction

^{1}are the only two so-called morphic numbers \(p \in \mathbb {R}\) greater than 1 that satisfy the following equations (Aarts et al. 2001):

*k*and \(l \in \mathbb{N}\). While the golden ratio is already known to mankind since antiquity and still is a subject of study in art, architecture (Xu et al. 2012; Huylebrouck and Labarque 2002) and various scientific domains such as mathematics, physics, biology and chemistry (Gonzlez 2010; Tung 2007; Yu et al. 2006; Swartzendruber et al. 1985; Boeyens and Thackeray 2014), the plastic number \(\psi\) was first defined and studied mathematically in 1924 by the engineer Gerard Codonnier. The first publication about this number, however, dates from 1960. That year, Dom Hans van der Laan

^{2}presented the plastic number through a new measurement scale of architectural proportions that is solely based on an empirical relation between the plastic number and the human capacity to visually perceive, distinguish and relate dimensions in threedimensional architectural space (Padovan 2002). As opposed to Le Corbusier’s well known Modulor, van der Laan’s measurement scale of architectural proportions does not take into account human dimensions, nor does it focus on the convenience of architectural spaces.

The main goal of this paper is to introduce the concept of so-called morphic angles, i.e. angles defined based on the morphic numbers. First, it will be explained how, based on the golden ratio, a morphic angle can be defined for all types of bivariate nondegenerated quadratic curves in two-dimensional Euclidean space. Subsequently, the concept of a morphic angle based on the plastic number will be introduced, defined and extensively discussed. Finally, it will be shown that through the concept of a morphic angle, a formula can be derived based on the plastic number that approaches the number \(\pi\) with good accuracy.

## The Circle-Based Definition of a Golden Angle

*a*and

*b*the length and the width of the golden rectangle, respectively. From the above relation, it can be derived that the golden ratio satisfies the following equations:

*r*that is sectioned according to the golden ratio (in literature this is often denoted as a golden section). This results in a so-called golden arc with length \(C_\varphi\) and an arc with length \(C^{'}=\varphi C_\varphi\) (see Fig. 1). Using Eq. (4), it can be determined that

A golden angle has been defined in two-dimensional Euclidean space as the central angle subtended by a golden arc and has a value of approximately 137.508\(^{\circ }\) (Prusinkiewicz and Lindenmayer 1990). To date, a golden angle has been studied in various scientific domains such as physics (Livio 2003) and medicine (Henein et al. 2011).

## Towards a Definition of a Golden Morphic Angle

### Introduction

^{3}

*e*the eccentricity and

*p*the focal parameter of a BNQC (see Table 1). The focal radius with length

*r*is the line segment between the considered focus point and a point on a BNQC. The true anomaly \(\nu _i\) at focus point \(P_i\) of a point

*P*on a BNQC is the angle measured between its focal radius and the minimal focal radius with length \(r_{\min }\) corresponding to a point \(P_{\min }\) on the BNQC closest to the focus point (see Fig. 2).

Eccentricity *e*, focal parameter *p* and minimum focal radius \(r_{\min }\) for each BNQC type

BNQC type | Eccentricity | Focal parameter | \(r_{\min }\) |
---|---|---|---|

Circle | 0 | \(\infty\) | |

Ellipse | ]0, 1[ | \(a(1-e^2)/e\) | \(a(1-e)\) |

Parabola | 1 | 2 | |

Hyperbola | \(]1,\infty [\) | \(a(e^2-1)/e\) | \(a(1+e)\) |

### Limitations of the Circle-Based Definition

If a golden morphic angle of a BNQC would be defined as the \(\Delta \nu\) related to a golden arc resulting from a golden section of the total circumference of a BNQC, three major problems would arise. Firstly, hyperbolas and parabolas are non-closed BNQCs, meaning that they do not have a finite circumference. Secondly, for ellipses, hyperbolas and parabolas, there is no exact formula to calculate the length of an arc segment, implying that the value of the corresponding \(\Delta \nu\) cannot be exactly determined. Finally, the value of \(\Delta \nu\) depends on the location of an arc segment with fixed length along the BNQC, implying that \(\Delta \nu\) does not have a unique value. It is thus clear that the circle-based definition of a golden angle cannot be applied for all BNQC types to define a golden morphic angle.

### Proposal of a New Definition

*ep*in Eq. (9) is not allowed for a circle because \(e=0\) and \(p=\infty\), implying that Eq. (11) cannot be mathematically derived for this particular BNQC. However, it can be easily verified that the value of the golden morphic angle obtained by applying basic goniometry (\(r'=r\), implying that \(sin\nu _\varphi =\frac{1}{\varphi }\)) is equal to the value obtained through Eq. (11) for \(e=0\). Moreover, it can be noticed that the morphic golden angle for a circle is part of a class of angles related to the concept of a golden triangle (i.e., an isosceles triangle with sides \(\varphi\), \(\varphi\), and 1). In this particular case, the angle is related to a so-called golden morphic triangle with sides \(\varphi\), \(\varphi\), and 2.

Approximate (range of) values of a golden morphic angle \(\theta _\varphi\) for each BNQC type

BNQC type | Golden morphic angle \(\theta _\varphi\) |
---|---|

Circle | ≈76.346° |

Ellipse | >76.346° and <126.871° |

Parabola | ≈126.871° |

Hyperbola | >126.871° and <180° |

## A Sphere-Based Definition of a Plastic Angle

*a*,

*b*, and

*c*the length, the width, and the height of the rectangular cuboid, respectively. From the above relation, it can be derived that the plastic number satisfies the following equation:

*R*and section its total area \(A_{s}\) in two areas \(A_\psi\) and \(A^{'}\) [see Eq. (4)] such that

## Towards a Definition of a Plastic Morphic Angle

### Introduction

A sphere can be considered as the simplest type of a nondegenerated real quadratic surface in three-dimensional Euclidean space that results from revolving a BNQC around an axis of symmetry going through the focus point(s). These surfaces will from now on be denoted as quadratic surfaces of revolution (QSR). There are four types of such surfaces: spheres, spheroids, paraboloids, and two-sheet hyperboloids.^{4} For each QSR, the cross-section perpendicular to the axis of symmetry results in a circle. Similar to the circle-based definition of golden angle, the solid angle considered in the sphere-based definition of a plastic angle can be regarded as the solid angle at the coinciding focus points of the revolved circle. Therefore, a solid angle at a focus point of the revolved BNQC will be considered in the search for a definition of a plastic morphic angle of a QSR.

### Limitations of the Proposed Sphere-Based Definition

If the proposed sphere-based definition of a plastic angle would be applied for each QSR type, a similar problem would arise as in the case of a golden angle (see Sect. 3.2): for paraboloids and two-sheet hyperboloids, the total area of the surface is infinite, meaning that it is impossible to realize a section based on the plastic number. It is thus clear that the sphere-based definition is not generally applicable.

### Proposal of a New Definition

*ep*in Eq. (22) is not allowed for a circle, implying that Eq. (24) cannot be mathematically derived for the related revolved BNQC. Again, it can be easily verified that the value of \(\nu _\psi\) obtained by applying basic goniometry (\(r'=r\), implying that \(sin \ \nu _\psi =\dfrac{1}{\sqrt{\psi }}\)) is equal to the value obtained through Eq. (24) for \(e=0\).

Approximate (range of) values of \(\nu _\psi\) and \(\Omega _\psi\) for each QSR type

QSR type | \(\nu _\psi\) | \(\Omega _\psi\) |
---|---|---|

Sphere | ≈60.344° | ≈3.174 |

Spheroid | >60.344° and <81.971° | >3.174 and <5.406 |

Paraboloid | ≈81.971° | ≈5.406 |

2-Sheet hyperboloid | >81.971° and <90° | >5.406 and <\(2\pi\) |

## A Plastic Approximation of \(\pi\)

*a*being the length of the smallest dimension of the plastic box. When comparing this with the area \(A_\psi\) of a plastic surface on a sphere with radius

*R*[see Eq. (16)], it can be seen that if \(a=R=1\), it holds that

## Conclusion

In this paper, the concept of morphic angles was introduced. Morphic angles were defined based on the morphic numbers for nondegenerated quadratic curves and surfaces of revolution. Notwithstanding that these angles were presented as new theoretical concepts, the author of this paper considers it possible that they have a deeper meaning in various scientific domains, especially in physics were the considered curves and surfaces are often encountered (particle motion, force fields, astrophysics, ...). Moreover, the author believes that the morphic angles defined in this paper may be of practical use in architectural design and construction.

## Footnotes

- 1.
The exact value of the plastic number is \(\dfrac{\root 3 \of {108+12\sqrt{69}} + \root 3 \of {108+12\sqrt{69}}}{6}\).

- 2.
Architect and member of the Benedictine order.

- 3.
For an ellipse, a circle and a hyperbola \(i \in \{1,2\}\), for a parabola \(i=1\).

- 4.
One-sheet hyperboloids will not be considered in this section, because their axis of symmetry is perpendicular to the axis of symmetry going through the focus points.

- 5.
A dihedral angle is an angle created by two intersecting planes.

- 6.
A great circle is the intersection of a sphere and a plane going through the sphere’s center.

- 7.
A regular plastic polygon has

*n*sides of equal length and*n*spherical angles of equal magnitude. - 8.
An oblique plastic triangle is a plastic triangle without right angles.

## Notes

### Acknowledgments

The author would like to thank professor Bernard De Baets (Research Unit Knowledge-based Systems, Ghent University, Belgium) and Thomas Van der Velde (School of Arts, Belgium) for their time and feedback. Any findings and conclusions are those of the author and do not necessarily reflect those of the above-mentioned persons.

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