In the world of mathematics, few figures loom as large as the ancient Greek geometer Euclid. Known for his foundational work in geometry, Euclid presented a set of five axioms that he believed to be self-evidently true. These axioms, or their equivalents, include the ability to draw lines between any two points, extend lines indefinitely, draw circles at any point with any radius, the equality of all right angles, and the fact that all triangles have internal angles that add up to 180 degrees. This system of geometry, known as Euclidean geometry, is what we learn in school and applies specifically to flat surfaces.
Exploring Curved Surfaces: The Mathematical Mysteries
However, when it comes to curved surfaces, such as the Earth, the rules of Euclidean geometry no longer apply. The internal angles of a triangle on a curved surface do not necessarily add up to 180 degrees, leading to a series of intriguing puzzles that challenge our understanding of geometry. Today, we delve into these mathematical mysteries to see if we can crack the code.
One such puzzle involves imagining the Earth as a perfect sphere. Can you draw two more identical lines from the North Pole to a point on the Equator to create a triangle where all the internal angles are right angles, totaling 270 degrees overall? This scenario pushes the boundaries of traditional geometric thought and encourages us to think outside the box when it comes to visualizing shapes on a global scale.
Challenging Conventions: Triangles on a Donut
Taking things a step further, let’s consider covering the Earth with equilateral triangles that have internal angles of 120 degrees, totaling 360 degrees overall. The task is to find a way to fit these triangles together seamlessly without any overlaps or gaps, creating a tessellation that spans the entire surface of the Earth. This challenge not only tests our geometric skills but also highlights the intricate relationship between shapes and space.
Now, shift your perspective to a donut, or torus, instead of a sphere. Can you draw two identical right-angled triangles on the donut so they perfectly cover its surface? What will the sum of the six internal angles of these two triangles add up to on this curved shape? This playful exercise in geometry invites us to explore the unique properties of different surfaces and how they influence the angles and shapes we encounter.
As we ponder these mathematical conundrums, it’s worth noting the rich history of mathematical inquiry that has shaped our understanding of geometry. Adam Kucharski, a mathematics professor at the London School of Tropical Medicine and a renowned science author, has delved into the intricacies of mathematical proof in his latest book, “Proof: The Uncertain Science of Certainty.” In this enlightening work, Kucharski explores how nineteenth-century thinkers challenged Euclid’s self-evident truths, paving the way for new mathematical insights across various disciplines.
Proof by Adam Kucharski is set to release in the UK this Thursday, offering readers a fascinating journey through the evolution of mathematical thought. From history and politics to statistics, computer science, and epidemiology, Kucharski’s book provides a comprehensive look at the ways in which mathematical certainty has been both questioned and reaffirmed throughout the ages.
As we await the release of Kucharski’s book, let’s continue to engage with the timeless puzzles and paradoxes that mathematics presents. The beauty of mathematics lies not only in its precision and logic but also in its ability to spark curiosity and creativity in those who dare to explore its boundless realms. So, let’s embrace the challenge, unravel the mysteries, and expand our minds through the captivating world of mathematical inquiry.
