The Entanglement of Everyday Maths in the Mysteries of Physics

By Sian Leo

Mathematics is often regarded as the universal language of the universe. From the simple calculations of daily life, to the complex equations governing the cosmos, maths serves as a bridge between the abstract and the tangible. Yet, its role in physics - especially when we delve into the mysteries of the universe - often reveals a far deeper connection. The so-called "entanglement" of everyday maths with the complexities of modern physics is not just a metaphorical idea; it is a concrete reality. Our daily lives are shaped by a multitude of basic mathematical principles, which are intimately connected to some of the most profound questions in physics.

The Hidden Geometry of the Universe

We use geometry in our everyday lives without necessarily thinking about it. Whether it’s calculating the area of a garden or determining how to fit furniture in a room, basic geometric principles take effect. However, the very same principles underpin some of the most advanced areas of physics.

Einstein’s General Theory of Relativity is a prime example of this. General relativity, which describes the force of gravity, is grounded in the concept of curved space-time. While this sounds like a concept that could only belong to the realm of abstract theory, it is rooted in the very real and everyday understanding of geometry. In the simplest terms, gravity is the result of the curvature of space-time caused by massive objects, such as planets. This idea, though profound, hinges on familiar concepts like the Pythagorean theorem and higher-dimensional geometry: ideas that are fundamentally geometric, yet applied to the very fabric of the cosmos.

The notion of space and time being curved around massive objects (like the Earth or the Sun) is not just confined to physics labs; it forms the very basis of GPS technology. The satellites that provide navigation services use Einstein's equations to account for the difference in time experienced at the satellite's altitude, compared to on Earth, due to relativistic effects. Everyday maths - such as calculating the time difference between two points - is essential for ensuring these systems work accurately. This perfectly explains time dilation, which refers to the seemingly odd fact that time passes at different rates for different observers, depending on their relative motion or positions in a gravitational field.

Probability and Quantum Mechanics

The odds of drawing a certain card from a deck. Predicting the likelihood of rain on a Monday morning. The chance of winning the lottery. Probability is a concept we encounter on a regular basis. In quantum mechanics however, probability takes on an entirely new level of significance.

At the heart of quantum mechanics lies the wave function, which describes the probabilistic nature of particles at microscopic scales. Rather than determining a particle’s precise location or velocity, quantum physics focuses on the probability distribution of where a particle might be found. This is governed by mathematical principles that are influenced by simple probability theories.

The Heisenberg Uncertainty Principle, one of the key concepts in quantum mechanics, is mathematically related to the fact that we cannot precisely measure certain pairs of variables (such as position and momentum) simultaneously. This uncertainty isn’t a limitation of technology, but a characteristic feature of nature, as suggested by the maths of probabilities.

Additionally, the famous concept of quantum entanglement - where particles become "linked" and instantaneously affect each other despite vast distances - relies on advanced mathematical descriptions of the wave function and linear algebra. Though these ideas seem detached from everyday maths, the foundation of their mathematical language lies in principles of probability and algebra.

Symmetry and Conservation Laws

Another deep connection between everyday mathematics and the mysteries of physics lies in the role of symmetry - a concept that we use to describe balance or proportionality in objects around us. A square, for instance, is symmetrical because it can be rotated by 90 degrees about its centre without changing its appearance. Similarly, the concept of symmetry plays a crucial role in the laws of physics.

In particle physics, the fundamental forces and particles of the universe are understood in terms of symmetry groups. These mathematical structures describe the lack of change in physical laws. For example, the well-known law of conservation of energy - that energy cannot be created or destroyed, only transformed - can be derived from the symmetry of time. This is because the laws of physics are the same at all moments in time, meaning energy must be conserved.

This kind of mathematical symmetry extends beyond theoretical physics into practical applications. Engineers regularly use symmetries in structural design to ensure that bridges and buildings remain balanced and stable. In fact, concepts from group theory, a branch of abstract algebra, are routinely used in the design of materials and the study of mechanical properties in engineering.

The Power of Linear Algebra in Physics

In response to many further mathematicians' perplexed inquiries regarding the uses of the seemingly random topics of vectors and matrices, linear algebra - a field of mathematics dealing with such subjects - is another area where everyday maths is entangled with the complexities of physics. By calculating displacement, velocity or forces in physics, basic vector arithmetic may seem straightforward; however, it’s these same tools that form the backbone of much of modern physics.

In quantum mechanics, linear algebra is used to describe the states of quantum systems. The quantum state of a system is often represented as a vector in a mathematical space, and the physical components (such as position, momentum, and energy) correspond to matrices that act on these vectors. More complex phenomena, such as quantum entanglement, quantum computing, and even the study of black holes, are deeply rooted in linear algebra. Don’t let that suck you in!

Moreover, relativity and electromagnetism - two of the most well-tested theories in physics - also rely heavily on linear algebra and matrix transformations. These mathematical structures allow physicists to manipulate complex systems and solve real-world problems that influence the behaviour of light, matter, and spacetime.

The Mathematical Foundation of Chaos Theory

Not all connections between everyday maths and physics involve neat, predictable relationships. In fact, many of the most puzzling systems in physics - such as weather patterns, turbulence in fluids, or the behaviour of certain particles - are determined by chaos theory. This field examines systems that, despite appearing to be random and unpredictable, are governed by deterministic rules.

The study of chaos theory often involves solving nonlinear differential equations (maths that may seem too abstract for most daily situations). Yet, the underlying concepts can be seen in everyday phenomena, from the oscillations of a clock, to the growth of populations. Even seemingly simple systems can produce complex and chaotic behaviour when initial conditions are slightly altered: a concept captured by the butterfly effect in chaos theory.

Understanding chaos has profound implications in both physics and mathematics. In the study of fluid dynamics, for example, chaos theory is used to model turbulence, which remains one of the most unmanageable problems in physics. Although the ability to predict such chaotic behaviour still has a long way to go, the fundamental mathematics involved has been key in uncovering a deeper understanding of complex systems.

Conclusion

The relationship between everyday maths and the mysteries of physics is not only fascinating, but crucial for our understanding of the universe. Through the geometry of curved space-time, the probabilities of quantum states, and the symmetries underlying physical laws, maths is the lens through which we understand both the simplest and most confounding phenomena. By recognizing the deep entanglement between the maths used in classrooms and the concepts that construct the laws of nature, we open the door to a deeper appreciation of the universe - one that is mathematically beautiful and infinitely complex. The universe may seem vast and intricate, but its underlying structure may be closer to home than you think.