Pythagoras twisted squares: Why did they not teach you any of this in school?
Mathologer・2 minutes read
The iconic Mathologer diagram illustrates Pythagoras' theorem through the combination of squares and triangles, with historical proof found in a Chinese manuscript. Various extensions of Pythagoras' theorem are explored through geometric means, leading to conclusions about bug movement in a square scenario.
Insights
- The Chinese manuscript on astronomy and mathematics contains the earliest documented proof of Pythagoras' theorem, featuring a second twisted square diagram, showcasing the historical significance of visual proofs in mathematical discoveries.
- The bugs in the scenario form a square due to a repeated movement pattern, resulting in an infinite number of squares with the bugs circling the center infinitely often, highlighting the concept of infinite patterns and the fascinating relationship between geometric shapes and movement.
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Recent questions
What is Pythagoras' theorem and how is it visually proven?
Pythagoras' theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This fundamental mathematical truth is visually proven by combining four right-angled triangles into a large square. The area of this large square equals the sum of the area of a smaller square and the four triangles, providing a clear geometric representation of the theorem.
What is the significance of Pythagorean triples?
Pythagorean triples are sets of three positive integers that satisfy Pythagoras' theorem, such as 3-4-5 where 3 squared plus 4 squared equals 5 squared. These triples have been historically important in mathematics and are illustrated within diagrams showcasing arithmetic progressions of squares. The existence of Pythagorean triples demonstrates the versatility and applicability of Pythagoras' theorem in various mathematical contexts.
Who proved the existence of arithmetic progressions among primes?
Ben Green and Terry Tao are mathematicians who proved the existence of arithmetic progressions of arbitrary length among prime numbers. They showcased a sequence of 5 primes with a common difference, highlighting the intricate relationships and patterns that can be found within the realm of prime numbers. This groundbreaking proof sheds light on the distribution and structure of prime numbers in mathematics.
What is the Trithagorean theorem for 60-degree triangles?
The Trithagorean theorem for 60-degree triangles introduces an equation where the sum of the areas of two squares equals the sum of the areas of two triangles, including the original triangle. This theorem extends the principles of Pythagoras' theorem to triangles with a 60-degree angle, showcasing the interconnectedness of geometric concepts and the versatility of mathematical theorems in different geometric configurations.
How is the Hexagorean theorem derived from the twisted square diagram?
The Hexagorean theorem is derived from the twisted square diagram by showcasing 12 white triangles on the left and 6 on the right. Dividing the Hexagorean theorem by 6 gives the Trithagorean theorem for 120-degree triangles, demonstrating the intricate relationships and patterns that can be uncovered through geometric diagrams and mathematical proofs. This theorem expands upon the foundational principles of Pythagoras' theorem to encompass a broader range of geometric shapes and angles.
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