The mathematics & geometry of 2025

2025 might appear to be an unremarkable year for many, yet it possesses a remarkable mathematical significance. Some websites have already published the fun fact that the number 2025 is divisible by the sum of its digits, which totals 9. A few have added that it is classified as a Harshad number. Additionally, 2025 is the square of 45 [45² or 45 x 45 = 2025], a number that is recognised as the 9th triangular number. The sum of the digits of 2025 is (2 + 0 + 2 + 5 = 9), which is a perfect square (3²).

Furthermore, 2025 can be represented as the sum of three perfect squares: It is the sum of three squares 40², 20² and 5² [(40×40) + (20×20) + (5×5) = 2025].

When dividing 2025 into the components 20 and 25, their combined total is 45, thereby connecting it to the Kaprekar property.

Then, 2025 is the sum of the cubes of digits from 1 to 9 (1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ = 2025) and adding the first 45 odd numbers gives 2025 (1 + 3 + 5 + ··· + 89).

The number 2025 has many factors, including 1, 3, 5, 9, 15, 25, 45, 75, 225, and 675.

Moreover, When 2025 is added to its reverse, 5202, the result is a palindrome, 7227.

Finally, this year represents the first square year since 1936 and will be the sole square year throughout the 21st century. The subsequent square year will occur in 2116, 92 years from now.

Geometry of 2025

What few have dealt with is the geometry of the number. One may explore tetrahedra with edge lengths 3, 3, 3, 3, 5, 5 as $3^{4}times{5}^{2}is2025$. It appears that there is no other incongruent tetrahedron with these six edge lengths where the two edges of length 5 do not share a vertex.

The crux of the matter lies in determining whether it is feasible for the 5-edges to be positioned opposite each other instead of being adjacent. 

If we pursue this configuration, each edge would be congruent to its opposite, resulting in all faces being congruent as well. It is a well-established fact that such a tetrahedron can exist if the congruent faces are acute triangles, or right triangles if one permits the degenerate case of a rectangle. However, it is important to note that 5 squared is greater than the sum of 3 squared and 3 squared, which implies that the congruent faces must be obtuse. Consequently, the proposed tetrahedron with the 5-edges positioned opposite would ultimately fail to maintain its structure.

A few video makers have demonstrated wonderfully how 2025 works in geometry. 

The initial visual demonstration presented in the following video is derived from independently identified visual proofs by J Barry Love, published in the March 1977 edition of Mathematics Magazine, on page 74, and by Alan L. Fry, featured in the January 1985 issue of the same magazine, on page 11.

The subsequent visual proof showcased in this video is based on a visual proof by Parames Laosinchai, which appeared in the December 2012 issue of Mathematics Magazine on page 360).