07-21-2026, 01:15 AM
One Pythagoras for All Dimensions
Summary
The article presents a beautiful and surprisingly simple generalization of the Pythagorean theorem to higher dimensions. It begins with the familiar observation that, in a rectangle, the sum of the squares of the four sides equals the sum of the squares of the two diagonals. Extending this idea to a three-dimensional box, the author shows that the sum of the squares of all its edges is equal to the sum of the squares of its four space diagonals.
The main result proves that this relationship holds for every finite-dimensional rectangular box (hypercube or hyperrectangle): in an n-dimensional box, the total of the squared lengths of all edges is exactly equal to the total of the squared lengths of all body diagonals. The proof relies on counting the number of edges and diagonals and applying the multidimensional distance formula, revealing that both sums simplify to the same expression. The article offers an elegant perspective on one of mathematics' most famous theorems, demonstrating how a familiar geometric fact naturally extends into any finite number of dimensions with remarkable simplicity.
ARTICLE [PDF]
Summary
The article presents a beautiful and surprisingly simple generalization of the Pythagorean theorem to higher dimensions. It begins with the familiar observation that, in a rectangle, the sum of the squares of the four sides equals the sum of the squares of the two diagonals. Extending this idea to a three-dimensional box, the author shows that the sum of the squares of all its edges is equal to the sum of the squares of its four space diagonals.
The main result proves that this relationship holds for every finite-dimensional rectangular box (hypercube or hyperrectangle): in an n-dimensional box, the total of the squared lengths of all edges is exactly equal to the total of the squared lengths of all body diagonals. The proof relies on counting the number of edges and diagonals and applying the multidimensional distance formula, revealing that both sums simplify to the same expression. The article offers an elegant perspective on one of mathematics' most famous theorems, demonstrating how a familiar geometric fact naturally extends into any finite number of dimensions with remarkable simplicity.
ARTICLE [PDF]
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│ KONSTANTINOS MICHAILIDIS │
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│ KONSTANTINOS MICHAILIDIS │
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