Amid COVID’s first wave, I had the privilege to join forces with Eric Weisstein and his team at Wolfram Research to create the History of Mathematics Project, a virtual interactive gallery highlighting physical artifacts that are important to the history of mathematics, for the National Museum of Mathematics (MoMath) in New York City. Most of my mandatory confinement at home was spent navigating through online collections from world-class museums, locating outstanding mathematical artifacts and creating interactive and computational explanations for them.

The Interactive Exhibits

The site contains more than 70 artifacts divided into virtual exhibits, grouped into nine general topic areas of mathematics: counting, arithmetic, algebra, the Pythagorean theorem, geometry, primes, π, polyhedra and mathematics education.

&#10005

Each of the nine virtual exhibits gives a brief textual description of a mathematical area and its historical context, together with navigation to other exhibits, clickable thumbnails for individual artifacts within the exhibit and a clickable timeline.

&#10005

Each artifact’s page contains a description of how the relic was used, detailed images, dimensions, location and additional suggested reading. Visitors can easily navigate to related artifacts and get a comprehensive historical overview in the timeline. Interactive content gives a hands-on exploration of each artifact, while computational explanations provide details about the mathematical content; both make extensive use of the Wolfram Language.

The Learning Journeys

After spending the first few months of the project populating artifact pages with interactive content and computational explanations, I moved on to creating eight Learning Journeys. These highly visual and hands-on computational essays connect the dots between different mathematical artifacts while providing engaging and fun learning journeys for middle-school students on up:

• Mathematical Beans and Knotted Strings: Counting Methods from the Moche Culture
• Balancing Ducks, Frogs and Grasshoppers: Weights and Measures in Ancient Mesopotamia
• Show Your Work!: Doing Math Homework on Clay Tablets, Papyri, Wax Tablets, Bamboo Strips and Birch Bark
• Squaring the Apsamikku Circle: The Search to Solve One of the Oldest Problems in Math
• Making Machines Fly: Overcoming the Square-Cube Law
• The Mathematics of a Masterpiece: Portrait of Luca Pacioli
• Ancient Right Triangles: The Pythagorean Theorem and the Gou-Gu Rule
• Ancient Games of Chance: The Beginnings of the Mathematical Theory of Probability

Learning Journeys are graphical, interactive computational essays that explore mathematical ideas across cultures and time. They are primarily visual and descriptive with minimal advanced mathematics, making them ideal for classroom exploration or as a teaching tool. At the same time, they contain images and links to more detailed mathematical artifact write-ups.

Ancient Games of Chance

Motivated by the ancient dice in the Polyhedra virtual exhibit, I embarked on a journey researching games of chance. I wanted to better understand their use throughout history and their connection to the development of the mathematical theory of probability.

I learned a lot of interesting things along the way. For example, a 1970s Italian expedition to the Burnt City (Shahr-e Sukhteh), located in the southeastern part of Iran, explored a graveyard in the ancient city and found a set of four elongated dice and 27 tokens, crafted about 4,500 years ago. This four-value die evolved into the modern six-sided cubical die.

And here are the probabilities for obtaining a given total using two six-sided dice:

&#10005

The product of my discoveries is an exploration into ancient games of chance. Did you know early dice were made of the knucklebones of sheep? Read the rest to find out more.

Balancing Ducks, Frogs and Grasshoppers

If you search for dice at the Penn Museum collections, it returns a staggering 749 items from around the world! When compared to other mathematical artifacts, dice are only outnumbered by scale weights. The Penn Museum has 2,591 historical weights on record, most of which come from the Near East. I dedicated an entire Learning Journey to these computational bits from the past.

For example, barley was so important to the ancient Mesopotamians that a barley grain was used as the smallest unit of length, area, volume and weight. A shekel of silver equaled 180 barley grains, or about 8.4 grams. Sixty shekels weighed one mina, and 60 mina weighed one talent.

In order to convince myself that this system was indeed accurate, I used the Wolfram Language to import, combine and visualize weight stones’ data from the ancient cities of Nippur and Ur:

&#10005

Here’s a data visualization of 628 weight stones from Nippur and Ur. The data was imported from two datasets curated and studied by archaeologist Brad Hafford. (The red lines indicate duck weights.) I was shocked by the resulting diagram: the majority of the weight stones clustered in weight values corresponded to well-defined fractions of a shekel or a mina!

&#10005

To understand how these weight stones might have been used, I created the following interactive balance scale️. As you add more barley grains on the left side of the scale, the merchant adds duck weights, which come in fractions of a shekel, to balance both sides. That means the beam at the top of the scale acts as an “equal” sign!

Learn More

If you are interested in knowing more about the making of a Learning Journey, join me at the Wolfram Virtual Technology Conference 2021. My Wolfram presentation “History of Mathematics Project: Learning Journeys for Kids and Others” is scheduled for Thursday, October 14, 11:30am–12pm CT US.

&#10005

Finally, let me invite you to the following free livestream events:

• “History of Mathematics: Debut of New Virtual Gallery with Wolfram CEO Stephen Wolfram,” October 27, 2021, 6–7pm ET
• “Making Machines Fly and Other Stories from Antiquity with Bernat Espigulé-Pons,” November 22, 2021, 6:30–8pm ET

The History of Mathematics Project, a virtual interactive gallery highlighting physical artifacts that are important to the history of mathematics and that is hosted at history-of-mathematics.org, was generously funded by Overdeck Family Foundation for the benefit of the National Museum of Mathematics (MoMath) in New York City. The virtual exhibits and their navigation were built entirely from source notebooks using the Wolfram Language, the Wolfram Knowledgebase and the Wolfram Cloud, which allow the content and website to be easily maintained and extended. Thank you to Heidi Kellner, Sarah Keim Williams, Lori Goodman, Andrea Gerlach and Eric Weisstein for all of your great contributions to the project!