The Ishango Bone: One of the earliest counting tools

(Image: Joeykentin, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons) 

The Ishango Bone was discovered in 1950 and is named for region of the Democratic Republic of Congo in which it was found. Carved into the bone are a series of tally marks. The exact use of the marks is not known, but it has been speculated they were for some mathematical operations or used as a lunar calendar. There is even the possibility one set of tallies on the stick represent the prime numbers between 10 and 20 (11, 13, 17, 19); however this could just be a coincidence. With an age of approximately 20,000 years, it is one of the earliest uses of humans using mathematics. 

With their historical significance, the tallies on the Ishango Bone are recorded in the On-line Encyclopedia of Integer Sequences as A200066 and A100000.

There is a similar artifact from Africa dating back 40,000 years called the Lebombo Bone.

Both artifacts fall under the general category of paleolithic tally sticks.

The Monster: 808017424794512875886459904961710757005754368000000000

(Image: https://www.iconfinder.com/Spot)

While reviewing entries in the On-line Encyclopedia of Integer Sequences, I came across one of the larger integers that is a term of a sequence: 808017424794512875886459904961710757005754368000000000. It is the order of the 26th Sporadic Simple Groups and is also known as the Monster Group.

Grant Sanderson has a video on the 3Blue1Green series: Group theory, abstraction, and the 196,833-dimensional monster. 

The OEIS sequence with this number is found here: https://oeis.org/A001228.

Other mathematicians posted videos about their favorite number over one million at: https://www.youtube.com/results?search_query=MegaFavNumbers.

Kissing Numbers

(Image: Tkdeason, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons)

The kissing number of a lattice structure is the number of other spheres that a sphere can touch. The maximal kissing number has been determined for dimensions up to 8. These kissing numbers are part of the On-line Encyclopedia of Integer Sequences as A001116 (0, 2, 6, 12, 24, 40, 72, 126, 240, 272).

For example, in the 2-D lattice shown above, the simple cubic arrangement has a kissing number of 4, while the hexagonal packing arrangement has a kissing number of 6 (the maximal number for 2-D). In a 3-D lattice, the maximal kissing number is 12

This blog has explored lattice structures in multiple dimensions in prior posts. See:

Atomic interstitial sizes in higher dimensions

Book Review – Instant Mathematics by Paul Parsons and Gail Dixon

Fields Medal Awarded to Maryna Sergiivna Viazovska (Марина Сергіївна Вязовська)

David Hilbert Problems

Book Review: Zero, The Biography of a Dangerous Idea by Charles Seife

(Image: https://www.ntskeptics.org/books/zero.gif)

Charles Seife, professor at New York University, wrote Zero, The Biography of a Dangerous Idea in 2000. Zero tells the story of how different cultures have used, or refused to recognize, the number zero in their mathematics. In telling this story, Seife also walks the reader through the history of mathematics including geometry, algebra, and calculus. A theme throughout the book is also the relationship of infinity and zero. In addition to the mathematical history, the book includes how zero and infinity link with many scientific concepts, including the future of the universe, absolute temperature, string theory, and quantum mechanics.