Monday Morning Math: the witch of Agnesi

Happy Halloween!  For this day, it seems appropriate to talk about the Witch of Agnesi. Oooohhhhh!  Spooky! 

But this witch is a curve.  It’s described with the algebraic formula:   (or, more generally, as – in the previous equation I used ).

The original construction is described geometrically, starting with a circle of radius .  Here’s a short video of how it is constructed

So how did this curve get its name?   Maria Gaetana Agnesi was a wicked smart woman who was born in Milan, Italy, in 1718, about 15 years after this curve was first studied by Pierre de Fermat and Guido Grandi.  By the time she entered her teens she spoke 7 languages, and by the time she entered her twenties she was also accomplished at philosophy and mathematics, which she discussed with her father’s visitors, part of an intellectual salon.  After her mother died Maria Agnesi took over running the house and wrote a calculus book for some of her 20 younger siblings.  This book was published when she was 30, when the study of Calculus itself was only decades old.  It was over one thousand pages, and Agnesi was granted an honorary appointment at the University of Bologna.  Agnesi spent most of her adult life focused on theology and serving others, particularly people who were poor or sick.

In 1901, John Colson translated it into English, and here is where a significant mistranslation occurred.  Agnesi had called the above curve averisera (related to the word for “turning”), but the word for “witch” is similar, avversiera, and that’s what Colson used. 

(Public domain from Wikimedia)

Sources:

• Kat Eschner’s article and Evelyn Lamb’s article in Smithsonian Magazine
• Frank J. Swetz and Victor J. Katz’s article in Mathematical Treasures
• Elif Unlu’s biography of Maria Agnesi

Monday Morning Math: Happy Divali!

Good morning!  Today is Diwali: for those unfamiliar with the day, it is a celebration of the victory of light over darkness that originated as a Hindu religious festival and is now observed by people in many religious traditions. (More details can be found in, for example, the Times of India.)  Many years ago I ran across an article discussing mathematics and Diwali and saved the link.  Alas, when I looked for it now the original article appears to be gone, but in doing a new search I found a few more recent posts:

First grade teachers Michelle L and Michelle M talked about patterns and symmetry with their students, using designs on rangoli.  Rangoli are intricate patterns made on the floor out of flowers, rice, sand, or other materials:

CC Subhashish Panigrahi

CC by Rajeev Nair

CC by Dinesh Korgaokar

One thing I find interesting about this is that the symmetry is different in each case – it’s not always 8 or 12 for example.  

And for those of you craving something a little more Calculus-oriented, take a look at this video:  

Enjoy the week!

Monday Morning Math: Tying a Tie

Good morning everyone!  Today we’ll learn about tying a tie.  This is a Windsor knot:

CC by Pumbaa80

But there are many other ways!  How many?  That’s the question that two physicists, Thomas Fink and Yong Mao, asked back in 1999.  They used math to prove that if you have a few rules, like that any tucks (folding one end of the tie under the rest) occur at the end of tying, then there are exactly 85 ways to tie a tie.  Not all of these are pretty, though: they thought only 13 of the ways really looked good.  Still, that’s a lot of neat ways, and they wrote a book called The 85 Ways to Tie a Tie: The Science and Aesthetics of Tie Knots.

But the story doesn’t end there!  Fifteen years later mathematician Mikael Vejdemo-Johansson saw a fancy tie on the Merovingian in The Matrix Reloaded that wasn’t in the list of 85 possibilities.  He loosened some of the rules that Fink and Mao had used and redid the math: with more options, he discovered a whopping 177,147 ways!  Here’s a knot called, appropriately, the Merovingian: 

CC by Pumbaa80

Using this new collection, if you tried a new way every hour, it would take you just over 20 years to get through all the possibilities.

More detail about the discovery, including a video of how to tie your tie, can be found in this 2014 article by Rose Eveleth in Smithsonian Magazine.

Monday Morning Math: Star Quilts

Happy Monday!  In honor of Indigenous Peoples’ Day our topic today is Star Quilts.  Here are some examples:

Here is the  Blue Day Star Quilt, probably of the A’aninin (Gros Ventre) in Montana, made in 1990-91.

Here’s the Fall Time Star (Bright Fall Day) Quilt by Marie Kinzel of the Lakota (Teton/Western Sioux) in South Dakota, made between 1968 and 1988. 

Created by Marie Kinzel (Marie Strong Heart Kinzel)

Below is the Earth and Sky Quilt created by Chantelle Blue Arm of Cheyenne River Lakota Sioux in South Dakota, made in 2014.  

Created by Chantelle Blue Arm

You can see many more examples at the National Museum of the American Indian.  I first read about them in the January 2021 issue of Crux Mathematicorum, published by the Canadian Mathematical Society. The article “Explorations in Indigenous Mathematics: No. 1” by Edward Doolittle begins:

The starblanket design is popular among the Indigenous peoples of the Plains region, particularly in quilted blanket designs, but also in other crafts. In the Plains Cree language, the word for star is atāhk and the word for blanket is akohp, so starblanket is atāhkakohp. Chief Ahtahkakoop, so named because “the stars blanketed the sky, more numerous and brighter than usual” the night he was born, was one of the first signatories of Treaty Six. Ahtahkakoop Cree Nation is named after the chief.

As explained in more depth in that article, these beautiful quilts lend themselves to math explorations. For example, and this will be familiar to anyone who had Problem Solving with me back in 2021:  How many tiles are there in a Star Quilt? (I added that the answer should have n in it, although each person needed to specify what n represented.) I remember enjoying reading these solutions, and I think people liked coming up with them, too.

Monday Morning Math: the First Digit Law

You can catch criminals with math!  You might expect that if you were to record a bunch of numbers, the first digit would be equally likely to be 1, 2, 3, 4, 5, 6, 7, 8, or 9, but it turns out that for many numbers (costs for a company, time spent working on something) the first digit is usually small:  1 is the first digit about 30% of the time, while 9 is the first digit less than 5% of the time! Here’s a picture of the distribution:

from Gknor 

This rule is generally known as the First Digit Law, although it is also called Benford’s Law after Frank Benford (who himself called it “The Law of Anomalous Numbers” in a 1938 paper) or the Newcomb-Benford Law in recognition that  Simon Newcomb had noted it more than 50 years earlier, in 1881, in “Note on the Frequency of Use of the Different Digits in Natural Numbers”.

There are also some restrictions on what kind of numbers follow the First Digit Law:  According to Statistics How To:

Benford’s law doesn’t apply to every set of numbers, but it usually applies to large sets of naturally occurring numbers with some connection like:

• Companies’ stock market values,
• Data found in texts — like the Reader’s Digest, or a copy of Newsweek.
• Demographic data, including state and city populations,
• Income tax data,
• Mathematical tables, like logarithms,
• River drainage rates,
• Scientific data.

The law usually doesn’t apply to data sets that have a stated minimum and maximum, like interest rates or hourly wages. If numbers are assigned, rather than naturally occurring, they will also not follow the law. Examples of assigned numbers include: zip codes, telephone numbers and Social Security numbers.

(TwoPi, in a discussion about this, mentioned that books of logarithm tables tend to be dirtier in the beginning than at the end, in a visual application of the law.) According to J. Carlton Collins in the Journal of Accounting the data set should be somewhat large, at least 500 entries ideally.  Still, it’s a pretty impressive rule, and one that doesn’t quite make intuitive sense to me.

So about catching criminals?  Forensic accountants use this rule to catch people who falsify invoices, because falsified data doesn’t usually follow this expected pattern.  Go math!