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.

200,000,000,000,000,000,000,000 Stars

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

I've seen several different estimations of the number of stars in the universe. Recently, The Conversation, an online independent news organization, published this article by Brian Jackson: How many stars are there in Space?

He estimates there are approximately 2 trillion galaxies with each having an average of 100 billion stars giving a total of 200,000,000,000,000,000,000,000 stars.

Fibonacci Day (11/23)

(Image: By Hans-Peter Postel - Own work, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=1739679)

A high-school math fan of mine recently posted a reminder that November 23 is Fibonacci Day (based on the beginning numbers of the Fibonacci sequence: 1 1 2 3, or 11/23). An extension of a special version of this date would be 11/23/5813 (using the next numbers of the sequence 5, 8, 13). This seems like a long time away, but if one includes the Jewish calendar (currently in year 5784), this date is just 29 years from now.

For date conversions: https://www.ocjewish.com/calendar/view/year.asp?tdate=08%2F07%2F2053.

Prior post: https://jamesmacmath.blogspot.com/2023/09/rosh-hashanah-5784.html.

Pi Day: https://jamesmacmath.blogspot.com/2021/03/pi-day-3-14-2021.html.

The Fibonacci numbers are sequence A000045 in the On-line Encyclopedia of Integer Sequences. Additionally, there are thousands of other sequences which are variations based on this core sequence.

The 9 Most Massive Numbers in Existence (Tia Ghose article in Live Science)

The website Live Science  recently published an article by Tia Ghose, The 9 most massive numbers in existence. It covers many subjects that have been discussed in this blog, including Graham's Number, the scale of the universe, and prime numbers. Below is a link to the article:

The 9 most massive numbers in existence

Count to 9,192,631,770 in One Second

Counting to a very large number, such as 9,192,631,770, in on second may seem like an impossible task, but it is done on a very regular basis by atomic clocks. The International System of Units defines one second as 9,192,631,770 vibrations of the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom.

More on atomic clocks can be found here: https://en.wikipedia.org/wiki/Atomic_clock.

Official time in the United States is maintained by the U.S. Naval Observatory. A link to the official time is below:

https://www.time.gov/

The Somos Sequences

 

The online magazine, Quantamagazine, recently had an article on special recursive sequences: https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/. The family of sequences I found most interesting was the Somos-k sequences. They are named after Michael Somos who discovered them in the 1980s.

For example, terms in the Somos-4 sequence are given by a(0)=1, a(1)=1, a(2)=1, a(3)=1 and for n>3,
a(n)= (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4). So, the Somos-4 sequence is: 1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209…

The interesting thing about this sequence is that all the terms are integers even though there is a divisor in the equation of a(n). This is true for all Somos-k sequences with k<8. Once k exceeds 8, the sequences start with integers but eventually yield non-integer values. With k=8, this occurs with 18^th term.

A Numberphile video speaks about this specific sequence, called the Troublemaker Number: https://www.youtube.com/watch?v=p-HN_ICaCyM.

Somos-related sequences in the Online Encyclopedia of Integer Sequences include:

Somos-4: https://oeis.org/A006720

Somos-5: https://oeis.org/A006721

Somos-6: https://oeis.org/A006722

Somos-7: https://oeis.org/A006723

Term at which n Somos-k sequence first becomes nonintegral (for k>7): https://oeis.org/A030127

Women in Mathematics

(Image: Hypatia by Jules Maurice Gaspard, public domain)

I recently re-read Instant Mathematics (see prior post:  https://jamesmacmath.blogspot.com/2022/02/book-review-instant-mathematics-by-paul.html) and the authors, Paul Parsons and Gail Dixon, included several brief biographies of famous women mathematicians. Below, I've summarized these from the book. If there is someone that the authors or I omitted, please comment and I'll add her to the list.

Hypatia: born around AD 350 in Alexandria, Egypt; died AD 415.

She was dedicated to preserving Greek mathematical and astronomical works. She wrote commentaries on Apollonius's work Conics and Diophantus's Arithmetica.

See also: https://en.wikipedia.org/wiki/Hypatia.

Sophie Germain: born in Paris 1776; died 1831.

As with many women of the time, she was barred from entry into universities, but she obtained lecture notes to study on her own. When corresponding with the mathematician Joseph Lous Lagrange, she used the pseudonym Monsieur Le Blanc.

She is best known for her theories of elasticity and number theory.

See: https://en.wikipedia.org/wiki/Sophie_Germain.

Ada Lovelace: born in London 1815; died 1852.

Lovelace worked with Charles Babbage, inventor of the first mechanical computer. She published how the analytical machine could be programmed and this is believed to be the first published computer algorithm.

See: https://en.wikipedia.org/wiki/Ada_Lovelace.

Florence Nightingale: born in Florence, Italy in 1820; died 1910.

Many people might remember Nightingale more for her work in medicine and improvement of santitary healthcare, but some consider her implementation of clear, descriptive graphics used in statistical analysis.

See: https://en.wikipedia.org/wiki/Florence_Nightingale.

Sofia Kovalevskaya: born in Moscow in 1850; died 1891.

With women banned from Russian universities, she obtained special permission to attend the university in Heidelberg. Eventually she became the first woman to achieve a full professorship in mathematics at a North European university. She is best known for her contributions to the theory of differential equations and mathematical analysis.

See: https://en.wikipedia.org/wiki/Sofya_Kovalevskaya.

Emmy Noether: born in Erlangen, Germany in 1882; died 1935.

Noether had the same challenges of overcoming sexism in her field, but she became on of only two female students at the University of Erlangen. Her contributions were in abstract algebraic structures such as groups, rings, modules, vector spaces and lattices. She studied under the guidance of David Hilbert and Felix Klein. Upon earning her PhD, she lecture at the university. In the 1930's she fled Nazi persecution and moved to the United States where she received a position at Bryn Mawr College.

See: Emmy Noether.

Joan Clarke: born in London in 1917; died 1996.

Clarke studied mathematics at Cambridge University but she was prevented from receiving her full degree because of her gender. She is best known for her code-breaking work with Alan Turing. They were eventually able to break the German codes in real time resulting in a great military advantage for England.

See: https://en.wikipedia.org/wiki/Joan_Clarke.

Olga Ladyzhenskaya: born Kologriv, Russia in 1922; died 2004.

She completed two PhDs at Moscow State University. Her contributions in partial differential equations was critical to the study of fluid mechanics, aerodynamics, and meteorology.

See: https://en.wikipedia.org/wiki/Olga_Ladyzhenskaya.

Katherine Johnson: born White Sulphur Springs, West Virginia in 1918: died 2020.

Many readers will recognize Johnson as her contributions to the NASA program was portrayed in the movie Hidden Figures in 2016. As part of NASA's Computing unit, she complete complex calculations for engineers. Early computers determined trajectories for orbital space flights, but astronaut John Glenn insisted that the computer's calculations were confirmed by hand by Johnson.

See: https://en.wikipedia.org/wiki/Katherine_Johnson.

Maryam Mirzakhani: born in Tehran in 1977; died 2017.

Mirzakhani is the first woman to receive the Fields Medal, the highest mathematical honor in the world. She advance algebraic geometry to understand the structure and complexities of curved spaces. Her PhD thesis solved a highly complex problem regarding the number of loops of a given length on a hyperbolic surface. This work has applications in string theory.

See: https://en.wikipedia.org/wiki/Maryam_Mirzakhani.

Karen Uhlenbeck: born in Cleveland, Ohio in 1942.

In 2019, Uhlenbeck was awarded the Abel Prize for "her pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems, and for the fundamental impact of her work on analysis, geometry, and mathematical physics." Her findings have helped physicists develop tools to understand the behavior of electromagnetic fields and quantum particles.

See: https://en.wikipedia.org/wiki/Karen_Uhlenbeck.

Other resources:

https://en.wikipedia.org/wiki/Category:20th-century_women_mathematicians

https://en.wikipedia.org/wiki/Category:21st-century_women_mathematicians

Book Review: Pluses and Minuses

 

Stefan Buijsman, philosopher of mathematics, earned his PhD at age 20. His book, Pluses and Minuses, How Math Solves Our Problems, covers several mathematical concepts in a very easy-to-read and understandable format. Many topics he reviews includes systems that we interact with everyday such as mapping the shortest route, Google searches, and Netflix movie recommendations. There are also examples of how mathematics models scientific observations and also predicts the possible existence of yet undiscovered phenomena. Most notable was Paul Dirac's prediction of the positron or anti-electron.

Another topic Buijsman writes is why early civilizations developed mathematics. Related to this topic, he also explains how certain indigenous cultures get by without having words for numbers. 

I particularly liked his chapter on probability and statistics. As an instructor of statistics myself, I will use Buijsman's examples in future classes.

Publisher link: https://www.penguinrandomhouse.com/books/611373/pluses-and-minuses-by-stefan-buijsman/

How Does a Math Formula Highlight the Creator?

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

The Ken Ham Blog recently had a thought-provoking post about the Golden Ratio: see link. It speaks how this ratio is related to many things in nature. A video by the Institute for Creation Research presents their argument for God as evidenced by intelligent design in the Fibonacci series and the Golden Ratio.

See also:

Speed Limit Conversions 

OEIS Sequence

42

(Image: iOS 7 icons icons by Visual Pharm (iconfinder.com))

In Douglas Adams' Hitchhiker Guide to the Galaxy, the answer to the ultimate question of the universe was 42. BigThink.com had an article today listing 5 actual questions of science that have the answer 42: 42 really is the answer to these 5 fundamental questions - Big Think.

More on Adams' classic: The Hitchhiker's Guide to the Galaxy - Wikipedia.

Update, 1/10/2024: Also see https://www.iflscience.com/life-the-universe-and-everything-why-42-really-is-the-ultimate-answer-72379 for additional questions answered with 42.

Book Review: Gödel, Escher, Bach: an Eternal Golden Braid

 

I recently re-read Gödel, Escher, Bach: an Eternal Golden Braid, an epic book weaving the similar patterns found in mathematics, art/design, and music. First published in 1979, just a few years after the author, Douglas Hofstadter earned his PhD from the University of Oregon in 1975. I first read the Pulitzer Prize-winning book in 1983 while I was earning my PhD from the University of Illinois. So, I found it very inspiring that someone could produce such a great work early in his career.

In addition to the math of Gödel, the designs of Escher, and the music of Bach, Hofstadter includes dialogues between Tortoise and Achilles in the style of Lewis Carrol to reinforce the ideas he puts forward. He also includes a number of puzzles, discussions on computer coding, recursion (on many levels), and most interesting, artificial intelligence. 

Wolfgang Haken, 1928 - 2022

 

(Image:  Wolfgang Haken in 2008 - own work)

One of my favorite math professors passed away last year and I regret not being timelier in reporting this. Wolfgang Haken, University of Illinois at Urbana- Champaign, together with Ken Appel, solved the four-color problem. This is the subject of an earlier post: Math Vacation: Four Color Map (jamesmacmath.blogspot.com). I really enjoyed his lectures and the enthusiasm he shared about mathematics.

 

Word Clouds

 

If you ever wondered how to produce a "Word Cloud," there is an easy function built into the program Wolfram Mathematica. I've been slowly taking tutorials on using Mathematica and a key to learn a new programming language is to apply newly introduced functions in your own work. 

If you don't have access to Mathematica, Wolfram Mathematica has a cloud-based version available for no cost - you just have to set up an account: see this link: Wolfram Cloud.

Next, copy the line of code below into a new notebook and replace "mathematics" with the subject of your choice.

WordCloud[WikipediaData["mathematics"]]

For example, to produce a word cloud for the city of Chicago, use: WordCloud[WikipediaData["Chicago"]].

The result is below.

More information specific to the WordCloud Function, see example 18 of: Strings and Text: Elementary Introduction to the Wolfram Language.

Tutorials on using Mathematica can be found here: An Elementary Introduction to the Wolfram Language by Stephen Wolfram.

Note: Stephen Wolfram is the author of A New Kind of Science. See: Math Vacation: My Favorite Math Websites (jamesmacmath.blogspot.com)

3,628,800

This post is inspired by a recent puzzle published by Alex Bellos in the Guardian: Did you solve it? Puzzles you can do in the pub | Mathematics | The Guardian

The title of the post is 10! = 1x2x3x4x5x6x7x8x9x10 = 3,628,800. Interestingly, this value is the same number of seconds in six weeks. The challenge in Bellos's puzzle was to make the comparison without using a calculator.

One can confirm this without a calculator. Write out the number of seconds in six weeks as:

6 (weeks) x 7 (days) x 24 (hours) x (60 minutes) x (60 seconds)

Now one can cancel factors in the above equation by matching up with terms in the equation for 10!.

Cancel the 6 in both equations.

Cancel the 7 in both equations.

Cancel the 24 by matching with the 3 and 8 in the 10! equation

Reduce the 60 to 6 in the seconds equation by cancelling the 10 in the 10! equation.

Reduce the other 60 to 6 in the seconds equation by canceling the the 2 and the 5 in the 10! equation.

These canceling operations leaves us with 6x6=36 in the seconds equations and 4x9=36 in the factorial equation, therefore the number of seconds in six weeks equals 10!.

Alex Bellos is also the author of one of my favorite math books: Math Vacation: My Favorite Math Websites (jamesmacmath.blogspot.com)

Slide Rule Collection Donated to Illinois Tech

Today, I received notice from the Library at the Illinois Institute of Technology that the slide rule collection I donated to the school is now on display. Over forty years ago, I started collecting slide rules and over the years, the collection grew to over thirty items. Earlier this year, I donated the collection to Illinois Tech, where I earned my Bachelor of Science degree in Materials Science and Engineering.

The collection also includes books of logarithmic and trigonometric values.

Some photos of the library display are shown below:

(photos by: Matt David, Director, Advancement Special Projects, IIT)

Other posts on slide rules: 

Math Vacation: 2022 Anniversaries (jamesmacmath.blogspot.com)

Math Vacation: Benford Distribution - Additional Thoughts (jamesmacmath.blogspot.com)

 

Anniversary of Euler's death

(Image: Jakob Emanuel Handmann - Kunstmuseum Basel, Public Domain)

Yesterday, September 18, 2023, was the 240th anniversary of the death of the Swiss mathematician Leonhard Euler. He was born in 1707 in Basel, Switzerland.

A more complete description of his life was recently published by Breakpoint Colson Center: Leonhard Euler: Called to Mathematics - Breakpoint (Includes an audio version).

Other posts in this blog related to Euler:

Math Vacation: Euler's Identity (jamesmacmath.blogspot.com)

(More on Euler’s Identity: Euler's identity - Wikipedia)

Math Vacation: Euler Characteristic of Polyhedra (jamesmacmath.blogspot.com)

Math Vacation: Project Euler (jamesmacmath.blogspot.com)

Math Vacation: Using Random Numbers to Estimate Pi (jamesmacmath.blogspot.com)

Math Vacation: Perfect Number Generator (jamesmacmath.blogspot.com)

More on Euler: Leonhard Euler - Wikipedia

 

 

Rosh Hashanah 5784

(Image: Iconfinder - ByteDance IconPark (oceanengine.com))

Last night (September 15, 2023) at sunset began Rosh Hashanah, beginning the new year 5784 in the Jewish Calendar. The Jewish calendar is the oldest calendar still in use today. It was started in the 9th century BC and based on biblical calculations that the creation took place in 3761 BC. 

Cinquante Signes

Éric ANGELINI writes the blog, Cinquante signes and is a frequent contributor to the Online Encyclopedia of Integer Sequences (OEIS). His blog provides further information on his sequences along with good graphics that illustrate the sequences.

A364831 and A364589: New Sequences Published in the Online Encyclopedia of Integer Sequences

I recently had two additional sequences published in the Online Encyclopedia of Integer Sequences.

A365589 - OEIS, numbers that have at least one prime digit and at least one nonprime digit and

A364831 - OEIS, primes whose digits are prime and in nonincreasing order.

In 1964 Neil Sloane started the maintenance of a list of integer sequences. Once his collection grew, he published the list in a book in 1973 ("A Handbook of Integer Sequences", by NJAS, Academic Press, NY). This book contained 2372 sequences.

Since 1996. the list in the form of a database has been maintained on the internet. As of the date of this post, it has over 300,000 different sequences that can be searched in many different ways. The formal name is The On-Line Encyclopedia of Integer Sequences or OEIS. This is a link to the database.

To view any of the sequences that I've authored or to which I have contributed, see: james c. mcmahon - OEIS.

An Open Message to the Blog's Fans in Singapore

(Image: Free 12 singapore icons - Iconfinder)

This past week, more views of this blog were made from Singapore than other country. To acknowledge these viewings, I wanted to write a post about Singaporean mathematicians. After a search, I found the following link on Wikipedia: Category:Singaporean mathematicians - Wikipedia. I'm asking my readers to add a comment to the blog in order to supplement this Wikipedia entry. For example, of these eleven mathematicians, are there a few who really stand out? Are there any people who should be added to the list? Thank you for your contributions!

• Louis Chen Hsiao Yun
• Chong Chi Tat

• Lam Lay Yong
• Peng Yee Lee
• Lek-Heng Lim

• Peng Tsu Ann

• Sun Yeneng

• Tan Eng Chye
• Tony Tan
• Kim-Chuan Toh

• Chen-Bo Zhu

10,000

My math blog just crossed over 10,000 views as of September 3, 2023, so I decided to write some interesting facts about 10,000.

Minnesota, also known as The Land of 10,000 Lakes, actually has 11,842 lakes.

For exercise, many people try to get in 10,000 steps in a day. That's approximately 5 miles or 8 kilometers.

Salmon P. Chase is featured on the U.S. $10,000 note.

There is a "$10,000 rule" which requires banks to report deposits or withdrawals of $10,000 or more of cash.

In English, myriad is used to describe a large number or a countless number of objects. The term is from the ancient Greek word for 10,000, myrias (μυριάς).

Update, 9/29/2023 - a $10,000 note recently sold for $480,000. Link to article.

Book Review: A New Kind of Science | Online by Stephen Wolfram

(Image: By Richard Ling - Own work; Location: Cod Hole, Great Barrier Reef, Australia, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=293495)

In 2002, Stephen Wolfram published A New Kind of Science (also known as NKS). It is based on the concept that relatively simple rules can create very complex systems. For the 20th anniversary, the book was made available free online: Stephen Wolfram: A New Kind of Science | Online—Table of Contents (wolframscience.com).

Before jumping in to read a 1000-page book on science, I suggest reading a few warm-up articles to help understand NKS:

John Conway's Game of Life: Conway's Game of Life - Wikipedia

and

Rule 30 - Wikipedia (one rule of many based on NKS that produces interesting results - perhaps the pattern shown in the photo of the shell shown above).

NKS touches on many different branches of science, including computing, randomness, physics, and nature.

Book Review - Infinite Powers by Steven Strogatz

(Image: Infinite Powers — Steven Strogatz)

Who would think that one could write a book about calculus without having it filled with endless equations? Steven Storgataz, professor at Cornell University, accomplished this task masterfully in his 2019 book, Infinite Powers, How Calculus Reveals the Secrets of the Universe.

I wish I had had his clear explanations of calculus over forty years ago when I was a young student of mathematics. His book is filled with clear explanations and illustrations. He also has many applications in which calculus has been critical, such as the disease research, fingerprint analysis, space exploration, and music.

Corporate Mission Statement Created by A.I.

 

(Image: emkamal kamaluddin on Iconfinder)

I recently ask Chat GPT to create a mission for an organization. Its response was as good as most organizations' statements (most bland at best):

Mission Statement for [Your Organization]:

"At [Your Organization], we are committed to empowering positive change and making a lasting impact in the lives of individuals and communities. Our mission is to inspire and uplift through innovation, compassion, and collaboration.

By fostering a culture of inclusivity and diversity, we strive to create a safe and nurturing environment where every voice is heard and respected. Through cutting-edge research, thoughtful advocacy, and sustainable practices, we aim to address pressing challenges and contribute to a better, more sustainable world.

Driven by our unwavering dedication to excellence and integrity, we seek to deliver exceptional products/services that enrich lives, inspire creativity, and promote well-being. We cherish our role as responsible global citizens and embrace opportunities to contribute to social progress and environmental stewardship.

Together, with our dedicated team, partners, and stakeholders, we work tirelessly to create meaningful solutions that transcend boundaries and create a positive ripple effect. We are guided by the belief that, through collective efforts, we can create a future where possibilities are endless and where every individual has the opportunity to thrive.

At [Your Organization], our purpose is not just to succeed but to leave a lasting legacy of positive change, shaping a world that reflects our core values and shared vision for a brighter, more compassionate tomorrow."

Note: many years ago, someone created the Dilbert Mission Statement Generator which I found produced a very believable mission statement about 1 in 3 times: Dilbert's Mission Stement Generator - WorldVillage.com - Where Culture & Technology Collide.

Contributor: (openai.com) Portions of this post labeled were written by ChatGPT, a language model developed by OpenAI. For more information on language models, visit the OpenAI website. Edited by: J. McMahon