The number (123…(b−1)) in base b has the property that when multiplied by any integer 1≤kb−1 which is coprime to b−1, its digits are permuted.
For example in base 10, 123456789
∗2 = 246913578
∗4 = 493827156
∗5 = 617283945
∗7 = 864197523
∗8 = 987654312

Discover more intriguing facts about both useless and useful numbers on our website: https://archimedes-lab.org/numbers/Num1_69.html

Two Meanings, One Word: The Tautology Tango

A tautology has two distinct meanings. In mathematical logic, it refers to a formula or assertion that is always true, regardless of the interpretation. For instance, “x=y or x≠y” is a tautology. Another example is the statement “either the ball is green, or the ball is not green,” which is always true, regardless of the color of the ball.

In everyday language, a tautology is a phrase that redundantly repeats the same idea in different words.

Some toponyms, which combine words from different languages, are often tautological. For example, Cheetwood (Lancashire) contains the words cę:to (Brittonic) + wudu (Old English), both with connotations of “wood, forest”. Similarly, Brill (Lincolnshire) is a combination of the Celtic word bre meaning “hill” and the English word “hill.” In the Pyrenees, the toponym Val d’Aran is a tautology, as “aran” means “valley” in Basque.

Montegibello in Sicily is another example, as it means “mountain mountain” in Latin and Arabic. In Algeria, the toponym Ain-Tala combines Arabic and Berber to mean “source source”. Other examples of tautological toponyms include Côtes-d’Armor in Brittany, which means “coast of coast” in French and Breton, and Dalsdalen in Norway, which means “valley’s valley” in Norwegian. Dasht-e Kavir in Iran means “desert desert”, while East Timor means “east east” in Indonesian and Malay. Minnehaha Falls in the US is named after the Dakota word for “waterfall”.

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Fibonacci Zoetropes

The Fibonacci Zoetropes are sculptures by John Edmark. The spirals in the sculptures follow the Fibonacci sequence. When filmed at 24 frames per second and spun at 550 revolutions per minute, each frame represents a 137.5 degree rotation, which is equivalent to the Golden Angle.

© John Edmark

Arithmetic Mean ≥ Geometric Mean

A simple yet neat visual proof demonstrating that the arithmetic mean of two positive numbers ‘a’ and ‘b’ is always greater than or equal to their geometric mean, symbolically represented as (a+b)/2 ≥ √ab

Amazing Double Cube

According to the Pythagorean theorem, adjacent cubes with side length 1 produce square roots of the first six natural numbers, as illustrated below:

From the book: Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, and Nature

Remarkably, by adding three extra cubes, we can extend the series of square roots of natural numbers up to √14. However, to obtain the square root of 7 using this method, we need to extend our analysis to a 4-dimensional world.

Harshad Years

Harshad number is defined as an integer that is divisible by the sum of its digits.

Interestingly, the years 2022-2025 are Harshad numbers. It is worth noting that having more than two consecutive Harshad years is a rare occurrence. The last time it occurred was over 1000 years ago for years 1014-1017. The next time it is expected to occur after 1000+ years will be during the years 3030-3033.