Pythagoras twisted squares: Why did they not teach you any of this in school?
Mathologer・25 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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Summary
00:00
Pythagorean Theorem: From Squares to Triangles
- The iconic diagram associated with Mathologer is commonly linked to Pythagoras' theorem.
- The fundamental mathematical truth of A squared plus B squared equals C squared is visually proven by combining four right-angled triangles into a large square.
- The area of the large square equals the sum of the area of the little square and the four triangles.
- The Chinese manuscript on astronomy and mathematics contains the earliest documented proof of Pythagoras' theorem, featuring a second twisted square diagram.
- The existence of Pythagorean triples like 3-4-5, where 3 squared plus 4 squared equals 5 squared, is illustrated within the diagram.
- The diagram showcases arithmetic progressions of squares, with Fermat determining that the maximum length is 3, aligning with Pythagorean triples.
- Ben Green and Terry Tao proved the existence of arithmetic progressions of arbitrary length among primes, showcasing a sequence of 5 primes with a common difference.
- Fermat's four square theorem, similar to his famous last theorem, remains a mystery in terms of the exact proof he claimed to possess.
- A Trithagorean theorem for 60-degree triangles is introduced, showcasing an equation where A area plus B area equals C area plus the area of the original triangle.
- Pythagoras' theorem extends beyond squares to equilateral triangles, where A area plus B area equals C area, even without the original triangle area.
15:30
"Geometric Theorems: 60 and 120 Degrees"
- 60 degrees is 90 minus 30, while 90 plus 30 equals 120.
- 60 degree triangles and 120 degree triangles have their own Trithagorean theorems.
- The Trithagorean theorem for 60 degree triangles involves A squared plus B squared equals C squared plus A times B.
- Eisenstein triples are the integer triples for 60 degree triangles, named after Gotthold Eisenstein.
- Equilateral triangles with integer sides, like 1,1,1 or 2,2,2, are examples of 60 degree triangles.
- There are 60 degree triangles with integer sides, such as the 3,8,7 triangle.
- The Hexagorean theorem is derived from the twisted square diagram, showing 12 white triangles on the left and 6 on the right.
- The Hexagorean theorem divided by 6 gives the Trithagorean theorem for 120-degree triangles.
- Various visual proofs of Pythagoras theorem are demonstrated using the twisted square diagram.
- The geometric mean of two positive numbers is always less or equal to their arithmetic mean, leading to the Cauchy-Schwarz inequality.
32:10
Bug's Infinite Square Path Length Analysis
- The bugs in the scenario form a square due to the repeated movement pattern, with each square shrinking in size by a fixed factor and rotating by a fixed angle. This results in an infinite number of squares, with the bugs circling the center infinitely often.
- The bug's path, although winding infinitely around the center, is of finite length. By calculating the lengths of the segments of the bug's path, it is determined that the total length of the path is approximately 1.14 times the length of one side of the square.
- Adjusting the fraction of sides covered by the bugs alters the length of the path, with the bug's path length approaching the length of the sides of the square. The bugs' paths in the original Martin Gardner problem are found to be exactly the length of the sides of the square.
