**Ah, architecture! **It truly has a rich history of incorporating mathematical patterns into building design.

The universe is full of energy, frequency, and vibration that make up everything around us. Nature is a true treasure chest of fascinating shapes and patterns, each one more beautiful than the last. These patterns are not just aesthetically pleasing, but they are mathematically intriguing. In this piece, we will delve into the relationship between shapes, mathematics, nature, and architecture.

One example of how mathematics and patterns relate in nature is the harmonic series. This series represents sound frequencies with different amplitudes. When plotted, the harmonic series takes on the appearance of a shell. While this comparison may seem odd, it's nothing more than a case of apophenia that plagues our minds when we seek patterns that may not exist. Fractals, however, are legitimate examples of mathematical patterns that occur in nature.

Shapes in higher dimensions are even more captivating. A circle in three dimensions, for instance, can be both a sphere and a spiral. What differentiates a three-dimensional circle from its two-dimensional counterpart is frequency and vibration. Spherical harmonics are wave functions of the electron in the hydrogen atom and serve as solid spherical shell oscillator functions. Our Greek ancestors could not have imagined such a picture!

In a Hilbert functional space orthonormal base, projecting a Euclidean space vector onto its Cartesian axis is no different from solving a manifold's harmonic equations in finite dimensions. This relationship holds true even for complex functions in the complex plane exp (i2πnt). We have versatile tools at our disposal to draw any shape we can imagine, even those beyond our wildest dreams.

The Laws of Mathematics are as rigid as human-made laws, but logic still reigns supreme in both fields. However, ambiguity can distort our reality on a perceived timeline. The resemblance between the Fibonacci sequence and the logarithmic spirals found in nature gives rise to speculation. It's worth contemplating whether galactic shapes inspire living math of that space. Is it also possible that all life forms emulate the kinetic galactic superwave? The relationship between geometry and harmonic functions has also been observed in architecture.

Architects have long used mathematics and geometric principles to develop aesthetically pleasing and structurally sound buildings. The Fibonacci sequence is one example of how geometric principles are applied in architecture. This pattern is a mathematical sequence that appears frequently in nature, such as in the arrangement of leaves on a stem or the spiral pattern of a seashell. Architects have incorporated this sequence into their designs, such as in the layout of the floor plan or the placement of windows.

This sequence consists of a specific series of numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…) where each value is the sum of the previous two. Nature also abounds in this pattern, from the spiral growth of snail shells to the arrangement of leaves on a plant stem. Architects use the Fibonacci sequence to establish beautiful proportions and spaces. For instance, the ratio of the height of the Parthenon's columns to their circumference is in the golden ratio, which comes from the Fibonacci sequence. The nautilus shell provides another example of a Fibonacci spiral, and this has been utilized to design things such as staircases and floor tiles.

Another set of patterns that architects have used in designing façades and building skins are fractals. Fractals exhibit repeating patterns that are self-similar at different scales. This means that one can find the same pattern at both micro and macro levels.Fractal geometry creates self-similar patterns that can be found in nature. These intricate and repeating patterns add harmony and balance to the built environment. They are often employed to produce intricate and visually stunning designs with repeating patterns, such as the façade of the Al Bahar Towers in Abu Dhabi.

The connection between shapes and mathematics extends beyond nature and architecture and into various fields such as music, art, and technology. The relationship between math and the world around us is an exciting and enthralling topic that continues to inspire scientists and thinkers alike.

In short, the beauty of shapes in nature and architecture is closely intertwined with the underlying mathematical principles that govern them. From the harmonics of sound frequencies to the patterns of the Fibonacci sequence, there is a profound link between math and the world around us that continues to captivate and inspire us. These patterns serve as a robust foundation for the design process, strengthening the fundamental relationship between mathematics, nature, and architecture.

It is therefore not a superficial connection but one that’s inherent in the mathematics that governs our universe. By exploring these connections, we may not know everything, but we can gain a more profound understanding of the world around us and how we as architects shape it.

Heini van Niekerk