Philosophies of the Soul — What Penrose Tiling May Teach Us About the Duality of Souls and Gender

What is Penrose Tiling?

Penrose tiling, named after the English mathematician and physicist Sir Roger Penrose, is a fascinating concept in the field of mathematics and theoretical physics. It involves a set of non-periodic tiling patterns that can cover an entire plane without repeating.

The author believes that Penrose tiling may also teach us something about the underlying structure of the universe, for example the duality found in many things, such as why there are predominantly two genders in sexual reproduction.

Historical Background

Sir Roger Penrose introduced Penrose tiling in the 1970s. His work on these tilings was part of a broader interest in the mathematical properties of tiling patterns and their implications for quasicrystals and the nature of space.

Sir Roger Penrose, image courtesy of Biswarup Ganguly

Types of Penrose Tilings

There are several different types of Penrose tilings, but the most famous ones are based on two different sets of tiles:

P1 (Kite and Dart)

• This tiling uses two shapes: a kite and a dart.
• The kite and dart tiles fit together in specific ways that enforce non-periodicity.

P2 (Rhombus or Penrose Rhomb)

• This uses two rhombus shapes, known as “thick” and “thin” rhombuses.
• The angles of these rhombuses are chosen such that they fit together without forming a repeating pattern.

P3 (Penrose Suns and Stars)

• This variation uses different shapes, sometimes resembling sun and star shapes.
• This set of tiles also maintains the aperiodic nature of the pattern.

Mathematical Properties

Penrose tilings exhibit several intriguing mathematical properties:

• Aperiodicity: Penrose tilings cover a plane without repeating patterns. This property is crucial because it demonstrates that non-periodic tiling patterns can fill an infinite plane without gaps or overlaps, unlike traditional periodic tiling patterns like those found in regular grid-based tilings.
• Local Isomorphism: While Penrose tilings do not repeat, any finite region of a Penrose tiling will appear somewhere else in the tiling. This property implies a form of local regularity within the overall non-periodic structure.
• Inflation and Deflation: Penrose tiles can be scaled up (inflation) or down (deflation) in a self-similar manner, where larger patterns can be divided into smaller tiles that follow the same rules.
• Decagonal Symmetry: The tilings often exhibit tenfold rotational symmetry. This property is significant in the study of quasicrystals, which possess rotational symmetries that are forbidden in periodic crystals.

Implications and Applications

Penrose tilings have had a significant impact on both mathematics and physics, especially in the study of quasicrystals:

• Quasicrystals: The discovery of quasicrystals in 1982 by Dan Shechtman, which exhibit non-repeating patterns similar to Penrose tilings, has shown that these mathematical constructs can describe physical structures. Quasicrystals have unique properties that differ from those of traditional crystals, such as unusual electrical and thermal conductivities.
• Theoretical Physics: Penrose tilings have implications in the study of space and symmetry, contributing to theoretical explorations in cosmology and general relativity.
• Art and Architecture: The aesthetic appeal of Penrose tilings has inspired designs in art, architecture, and even puzzles. The non-repetitive yet structured patterns offer a visually intriguing alternative to conventional tiling methods.

Penrose Tilings from Higher Dimensions

5D to 2D Projection

Penrose tilings can be derived from the projection of a five-dimensional lattice onto a two-dimensional plane. The process involves selecting a suitable “slice” of the higher-dimensional lattice and projecting it onto the plane in such a way that the resulting pattern maintains certain symmetries and aperiodic properties.

Lattice Points in 5D

• Start with a regular lattice in five-dimensional space, where the points are evenly spaced in all dimensions.
• Each point in this lattice can be described by five coordinates.

Projection Plane

• Define a two-dimensional plane in the five-dimensional space onto which the lattice points will be projected.
• The choice of the plane and the method of projection are crucial to ensuring the resulting pattern has the desired properties.

Selection of Points

• Not all points from the 5D lattice will be projected; a selection criterion, often involving “windows” or “acceptance domains,” is used to choose which points will map to the 2D plane.
• The chosen points are then projected onto the 2D plane, creating a pattern.

Resulting Pattern

• The projection results in a pattern that exhibits aperiodic order. The positions of the projected points correspond to the vertices of the Penrose tiling.
• The tiling obtained from this projection retains the non-repeating nature and local symmetry characteristics of Penrose tilings.

First Conclusion

Penrose tiling represents a beautiful intersection of mathematics, physics, and art. Sir Roger Penrose’s work on these non-periodic tiling patterns has provided deep insights into the nature of symmetry, aperiodicity, and the mathematical structures underlying physical reality. The study of Penrose tilings continues to be a rich field for mathematical exploration and practical application.

Speculation that there Might Be Parallels Between the Mathematics of Penrose Tiling and the Underlying Duality found in much of the Universe

The idea that higher-dimensional structures, such as those posited in Penrose tiling and the Kabbalistic Tree of Life, could relate to fundamental aspects of the universe, including duality and gender, is a fascinating intersection of mathematics, mysticism, and theoretical physics.

While these areas traditionally occupy different realms of thought, exploring potential parallels can provide intriguing insights. Here, we delve into whether the concept of five dimensions might relate to the duality of gender and the need for sexual reproduction.

Five-Dimensional Structures and Duality

Penrose Tiling and Higher Dimensions

Penrose tiling’s reliance on projections from higher-dimensional spaces to achieve non-periodic but ordered patterns suggests that higher-dimensional constructs can underpin complex structures observed in lower dimensions. This projection from five dimensions to two dimensions creates intricate, aperiodic tiling patterns that exhibit local symmetry and global aperiodicity.

Kabbalistic Tree of Life

The Kabbalistic Tree of Life, often interpreted through a mystical and spiritual lens, is described as having dimensions beyond the physical. In Kabbalah:

• Three Spatial Dimensions: Represent the physical aspects of reality.
• One Temporal Dimension: Represents time.
• One Spiritual Dimension: Represents higher, non-physical realities. The Tree of Life also encompasses concepts of duality and balance, with elements like masculine and feminine principles, which are seen as fundamental to the universe’s structure.

Parallels and Theoretical Connections

Duality in Physics and Cosmology

In theoretical physics, duality often refers to pairs of theories that describe the same phenomena from different perspectives. String theory, for example, uses higher-dimensional spaces (up to 11 dimensions in M-theory) to explain fundamental forces and particles. Dualities in these theories sometimes manifest as pairs of complementary descriptions (e.g., wave-particle duality).

Gender and Reproduction

In biological terms, gender differentiation and sexual reproduction are mechanisms that promote genetic diversity, enhancing the adaptability and survival of species. The existence of two genders (male and female) in most sexually reproducing organisms can be seen as a form of duality, crucial for the perpetuation of life.

Speculative Theories and Mystical Interpretations

Five Dimensions and Duality

If we consider the hypothesis that five-dimensional structures give rise to fundamental dualities:

1. Mathematical and Physical Dualities: Higher-dimensional constructs, like those seen in Penrose tiling, could imply that the inherent properties of these dimensions produce dualistic patterns when projected into lower dimensions. These dualities might include physical phenomena and perhaps more abstract concepts like gender.
2. Mystical and Spiritual Dimensions: The Kabbalistic view integrates spiritual dimensions into the fabric of reality, suggesting that dualities like gender may have deeper, spiritual roots. The interplay of masculine and feminine principles in Kabbalah reflects a balance that is essential for the holistic understanding of the universe.

Gender and Higher Dimensions

While mainstream science does not directly link higher-dimensional physics with gender, speculative theories could propose that the structures governing the universe at a fundamental level might also underpin the dualities we observe in nature, including gender:

• Energetic Duality: Some mystical traditions suggest that masculine and feminine energies are fundamental dualities that permeate all levels of existence. If higher-dimensional spaces influence lower-dimensional structures, these energies might be seen as manifestations of higher-dimensional dualities.
• Reproduction and Balance: The necessity of gender for sexual reproduction could be viewed as a reflection of a deeper, inherent duality in the universe, possibly originating from complex higher-dimensional interactions that ensure diversity and adaptability in life forms.

Second Conclusion

While there is no scientific consensus linking the five-dimensional structures used in Penrose tiling with the gendered aspects of the Kabbalistic Tree of Life, exploring these connections can provide a rich field for speculative thought. The idea that higher-dimensional spaces might give rise to dualities observed in our universe, including gender and the need for sexual reproduction, invites a multidisciplinary approach that spans mathematics, physics, biology, and mysticism. Such explorations remind us that the quest to understand the universe often bridges the boundaries between science and spirituality, encouraging us to consider deeper, perhaps hidden, dimensions of existence.