Monday Morning Math: Dorothy Vaughn

November 29, 2021 by

Dorothy Vaughn was born on September 20, 1910 in Kansas City. Missouri. She graduated at the age of 19 from Wilberforce University, a historically black college in Wilberforce, Ohio. Dorothy Vaughn supported her family as a math teacher for 14 years prior to working at NASA as part of the National Advisory Committee for Aeronautics’ (NACA) West Area Computing unit in 1943.

The West Area Computing unit was a group of black women who, as a result of Jim Crow Laws, were segregated at NASA while they performed mathematical calculations on slide rules and graph paper to support the space race and the NASA astronauts’ flight missions to space. Dorothy Vaughn was an expert in the computer programming language FORTRAN and she became NASA’s first black supervisor of the group in 1949 where she taught the women programming to prepare them for the future which she believed would be machine computers.

In addition to her work at NASA, Dorothy Vaughan raised her family of 6 children, one of whom also worked for NASA.

She retired from NASA in 1971 and died on November 10, 2008. She was featured in Margot Lee Shetterly’s book Hidden Figures and portrayed in the film based on the book by Octavia Spencer.

Sources:
https://www.nasa.gov/content/dorothy-vaughan-biography
https://scientificwomen.net/women/vaughan-dorothy-103
https://blog.sciencemuseum.org.uk/nasas-overlooked-star/

Written by Tracy Lyn Lause

Monday Morning Math: Hippocrates of Chios

November 15, 2021 by

No, not that one!

Hippocrates of Chios was a Greek mathematician who followed Pythagoras and lived around 470-410 BCE. He was a contemporary of the more famous Hippocrates, the physician Hippocrates of Kos (c460-370 BCE). He is perhaps best known for his work on the classical geometry problems of squaring the circle and doubling the cube.

Astute readers will note that both of the geometry problems above are impossible to solve in general, but Hippocrates’ work led him to discover how to compute the areas of certain lunes, regions bounded by two circular arcs. (See this video for a pleasing application of Hippocrates’ discovery.)

The “Lune of Hippocrates”

The Greek philosopher Proclus credits Hippocrates with writing the first version of Elements of Geometry, much of which Euclid would later incorporate into his own Elements. In this text, Hippocrates introduced the use of letters to represent points, as well as naming objects using the points that defined them (e.g., “triangle ABC”). It is also believed that he had developed a method of “proof by exhaustion” (approximating circles by polygons with an increasing number of sides), later used by Eudoxus and Euclid. Unfortunately, Hippocrates’ text has been lost to history.

While respected as a mathematician, Hippocrates was viewed as “stupid and incompetent in the business of ordinary life” by Aristotle. He lost a large sum of money due to fraud, and had to teach geometry in Athens to make up for it.

Sources:

https://en.wikipedia.org/wiki/Lune_of_Hippocrates
https://mathshistory.st-andrews.ac.uk/Biographies/Hippocrates/

Monday Morning Math: Grace Murray Hopper

November 8, 2021 by

In honor of Veterans Day, our mathematician this week is Grace Murray Hopper.

Grace Murray was born in New York in 1906. She earned her BA in math and physics from Vassar College. Over the next few years she married, earned an MA and PhD in math from Yale University, and became a professor at Vassar.

In 1943, at the age of 37, she joined the Navy in response to World War II and began working with computers. She worked on the Harvard Mark I (which was over 50 feet long, 8 feet tall, and 2 feet deep) and later the Mark II and III. She learned programming and was instrumental in both conceptualizing and creating the first compiler.

At the time of her retirement in 1986 she was at the rank of Rear Admiral and the oldest active military officer. She continued to work even after her retirement and died in 1992. She is buried in Arlington Cemetery.

One of her legacies is the popularization of the term computer bug. She invented the term “debugging” in response to an actual bug (shown in the photo below!)

Hopper found the first computer “bug” a dead moth that had gotten into the Mark I [possibly Mark II] and whose wings were blocking the reading of the holes in the paper tape. The word “bug” had been used to describe a defect since at least 1889 but Hopper is credited with coining the word \debugging” to describe the work to eliminate program faults.

(From computersciencelab.com)
Courtesy of the Naval Surface Warfare Center, Dahlgren, VA., 1988, public domain

Sources:

Monday Morning Math: Number Systems

November 1, 2021 by

For today’s Monday Morning Math, we’ll do some counting!

In the Indo Arabic number system we have ten digits: 0123456789 grouped by tens, so the number 23 can be thought of as 2 tens and 3 ones.  But in Mesopotamia (modern day Iraq) 4000 years ago they grouped numbers by 60s.  In that system the number 2 3 would be 2 60s and 3 ones, or what we would think of as 123.  That grouping by 60s, incidentally, has passed down to us in how we count time: it is the reason we have 60 seconds in a minute and 60 minutes in an hour.

For an example closer to home, the Mayans 2000 years ago, in modern-day Mexico, Guatemala, Belize and Honduras, grouped numbers by 20s.  In this number system the number 2 3 would be 2 20s and 3 ones, or what we could think of as 43. Numbers below numbers less than 20 were formed using dots for 1s and lines for 5s, with a shell like oval for the number 0, as in the image below.

Some Mayan numbers in the thousand year old Dresden Codex, named after where it was taken away to.

Several indigenous tribes in North American group numbers by 4s and 8s, particularly the Yuki in northern California, and the Chumash people along the central and southern coast.

(From “Numeral Systems of the Languages of California” by Roland B. Dixon and A. L. Kroeber, published in American Anthropologist in Oct-Nov 1907 (p. 665)

Grouping numbers by 5s and 10s is often attributed to fingers on the hand, but the interesting (to me) thing is that grouping numbers by 4s and 8s is also based on the hands:  the Yuki used the spaces between the fingers to count.  As Dixon and Kroeber wrote above, “It does not follow that because people count by their fingers they count by fives.”

November 1 marks the beginning of National Native American Heritage Month!  Visit the website for Indigenous Mathematicians to learn about more mathematicians!

 I first read about the Yuki in Ethnomathematics: A Multicultural View of Mathematical Ideas by Marcia Ascher. 

Monday Morning Math: Adolphe Quetelet

October 25, 2021 by

Adolphe Quetelet (pronounced Ket-eh-lay) was a Flemish Scientist who was the first to use the normal curve.

Born in Ghent, France in1796, Adolphe’s father died when he was just seven years old. At the age of 17, after his own schooling at Lyceum in Ghent where he excelled in mathematics, he took a job teaching mathematics at a school in 1813 to support his family. He was appointed a mathematics instructor at the College in Ghent in 1815 at the age of 19.

While at the College of Ghent, Adolphe was influenced by Garnier who encouraged Quetelet’s deeper studies in mathematics. He went on to earn a doctorate from the University of Ghent in 1819 with a dissertation on conic sections. After graduating and at the age of 23, he was appointed chair of elementary mathematics at the Athenaeum in Brussels. While he taught mathematics, Quetelet had a strong interest in astrology and lobbied for an observatory in Brussels. While visiting Paris on a fact-finding mission for the observatory, Quetelet learned the importance of statistical methods in astronomy.

As a result of his “zeal for statistics,” Quetelet identified society as a topic and studied and wrote papers on social statistics and in the course of that work was the first to use the normal curve/distribution and used what astronomers knew as the error law or bell curve on human populations. He also introduced the height/weight measure that we know today as the body mass index (BMI). He used the idea of an average as a central value. He collected statistics on crime and mortality and improved census taking for the government.

In 1855 Quetelet suffered a moderate stroke and never fully recovered suffering from a poor memory which negatively impacted his writings. Quetelet died in 1874.

Sources:
https://mathshistory.st-andrews.ac.uk/Biographies/Quetelet/
https://www.famousscientists.org/adolphe-quetelet/

Compiled by Tracy Lyn Lause

Monday Morning Math: Florence Nightingale

October 18, 2021 by

The mathematician this week is Florence Nightingale (1820-1910). WHAT??? Yes, she is most known for being a nurse, but many do not realize that Florence Nightingale was also a mathematician and data collector and is known to be one of the most prominent statisticians in history. Quite something for a woman in the 1850s.

Born in 1820 to wealthy parents, as a child, her father fostered her education in history, philosophy and literature, but she was gifted in math and the languages. Florence Nightingale was grounded in her religious beliefs and said she was “called by God” to “reduce human suffering” and as such pursued a career in nursing.

It was as a nurse during the Crimean War (1853-1856) in the Scutari, Turkey Barrack Hospital in 1856 that she collected data on the care of the wounded soldiers, recording information about how the soldiers died and “bringing order and method to the hospital’s statistical records.” The analysis of this data prompted Florence Nightingale to demand better care and more food and supplies for the soldiers in the hospital as well as the implementation of better hygiene and cleaning procedures. Florence Nightingale used her data to show the need for standards of care and “her accomplishments reduced the mortality rate to about 2 percent.”

Thus, it is said “her work in statistics saved lives.” She also earned the title “the Lady with the Lamp” from the soldiers for whom she cared since she made rounds in the evenings carrying a lamp through the hallways.

Florence Nightingale is credited with being an innovator in displaying statistical data through graphs (infographics). She uses a Coxcomb graph or Rose Chart (similar to a pie chart) in 1858 to illustrate the improvements to the mortality rate of soldiers in the hospital after her cleaning and sanitation procedures were adopted. Two years after returning from Crimea, Florence Nightingale was elected the first female member of the Statistical Society in 1858.

Sources:
https://www.britannica.com/biography/Florence-Nightingale
https://www.history.com/topics/womens-history/florence-nightingale-1
https://thisisstatistics.org/florence-nightingale-the-lady-with-the-data/
https://www.sciencemuseum.org.uk/objects-and-stories/florence-nightingale-pioneer-statistician

This post was generously written by our own Tracy Lyn Lause. Thanks Tracy!

Monday Morning Math: Alan Turing

October 15, 2021 by

(Monday morning Math – this week on Friday because Hello Midterms!)

Our mathematician this week is Alan Turning.  He was born in London, England, on June 23, 1912 and studied mathematics at the University of Cambridge.  After his graduation he wrote a paper called “On Computable Numbers, with an Application to the Entscheidungsproblem” which showed the depressing sounding but powerful result that not every true statement is provable in a mathematical system.  During this time he also invented the Turing machine, which is an abstract computer (as opposed to an actual physical computer) that performs logical computations.

Turing earned his PhD at Princeton University in 1938 and returned to Cambridge where he began working on codebreaking.  The following year, at the start of World War II, he moved to Bletchley Park where he developed methods for breaking various codes intercepted from the Germans and, even after the war continued to make significant contributions to work in artificial intelligence.

While Turing was recognized for his work he was also persecuted because of his sexuality:  in 1952 he was convicted of being homosexual and given the choice of going to prison or taking hormones as a chemical castration. He chose the latter; he also lost his security clearance as a result of this conviction.  Turning continued to do work in physics and biology, but died on June 7, 1954, from self-induced cyanide poisoning.  In 2009 the British government officially apologized for how they treated him and in 2013 Turing was posthumously pardoned. 

image.png

(This image of Alan Turning was made by Stephen Kettle from Welsh slate is sharable under creative commons, whereas the photos I could find still have a copyright.)

You can find more information about Alan Turing at BritannicaWikipedia and turing.org.uk  And to end on a more positive note, these days there is Spectra, the Association for LGBTQ+ mathematicians and their allies.

Monday Morning Math: Whose Triangle?

October 4, 2021 by

Today’s snippet isn’t a who, or at least not a single who.  There is a triangle that many people learn about in school, since it pops up in some interesting places.  It starts off like this

image.png

with each number equal to the the sum of the two numbers above it.  It’s a pretty interesting triangle, but who first came up with it?

In the United States it is often referred to as Pascal’s triangle, after Blaise Pascal.  This isn’t so much because he invented it (he didn’t) but because he wrote so much about it, in a book entitled Treatise on the Arithmetical Triangle (but in French, so Traité du triangle arithmétique), written in 1654 and published 11 years later.  Here’s how he wrote it:

image.png

But the Triangle was known before that.  Here’s a picture from 100 years earlier, in  a book by Niccolò Tartaglia in Italy:

image.png

Here’s one from 250 years before THAT, by Zhu Shijie in his book Si Yuan Yu Jian from 1303 in China:

image.png

There’s another version around this same time period by Omar Khayyam in Persia (modern day Iran) although I didn’t have a picture of that to include. But he wasn’t the earliest either: here’s one from 550 years before THAT (so roughly 900 years before Pascal)

image.png

This is from a manuscript in Raghunath Library, Jammu and Kashmir, in India from 755 (according to Wikipedia) where the figure was called the Staircase of Mount Meru (Meru-prastaara).  This, too, is unlikely to be the earliest: there are indications that the earliest manuscripts showing the arithmetic triangle are copied from even earlier ones.

So who first described this Arithmetic Triangle?  We don’t know, although we can say with assurance that it was well over a thousand years ago.

More information can be found at wikipedia, at  britannica.com 

Monday Morning Math: Ada Lovelace

September 27, 2021 by

Our Mathematician this week is Augusta Ada Byron, also known as Ada Lovelace.  She was born in London, England, on December 10, 1815, the daughter of the mathematically-inclined Anne Isabelle Milbanke and the poet George Gordon Byron (known more commonly as Lord Byron).  Her parents separated when she was a baby, and she was raised by her mother, who encouraged her in mathematics:

Lady Byron wished her daughter to be unlike her poetical father, and she saw to it that Ada received tutoring in mathematics and music, as disciplines to counter dangerous poetic tendencies. But Ada’s complex inheritance became apparent as early as 1828, when she produced the design for a flying machine. It was mathematics that gave her life its wings.

From ScienceWomen

When Ada was 17 she met Charles Babbage at a party, and he talked about his Difference Machine, a (very) early version of a computer.  Ada and Charles exchanged letters for nearly 20 years, throughout Ada’s marriage to William King, the Earl of Lovelace, and the birth of her children Byron, Anne Isabella, and Ralph Gordon.  After  a mathematician Luigi Federico Menabrea published an article about another of Babbage’s inventions, the Analytic machine, Ada translated the article from French to English, including notes of her own that were longer than the original article:

Her translation, along with her notes, was published in 1843, and represent her greatest contribution to computer science: she described with clarity how Babbage’s device would work, illuminating its foundations in the Jacquard loom. Just as Joseph-Marie Jacquard’s silk-weaving machine could automatically create images using a chain of punched cards, so too could Babbage’s system—the engine, Lovelace explained, weaved algebraic patterns. She also wrote how it might perform a particular calculation: Note G, as it is known, set out a detailed plan for the punched cards to weave a long sequence of Bernoulli numbers, and is considered to be the first computer program. 

From The New Yorker

Ada died of cancer on November 27, 1852, when she was only 37 years old, and the computer language Ada is named in her honor

Like podcasts?  Then I recommend this episode about Ada Lovelace from The History Chicks.

Like biographies?  During Hispanic Heritage month the website Lathisms is publishing a biography every day of a Hispanic/Latinx mathematician.  Lathisms was founded in 2016 by Alexander Diaz-Lopez, Pamela E. Harris, Prieto-Langarica, and Gabriel Sosa.

Monday Morning Math: Alberto Pedro Calderón

September 20, 2021 by

Welcome to the inaugural edition of Monday Morning Math! Every Monday Morning during the semester we’ll be posting some information about a mathematician, or some fun math. (For us it is, admittedly, a nice way to say Hello! to the blog again, which has not had many posts lately [*cough* understatement *cough*].

Our first mathematician is Alberto Pedro Calderón.  He was born on September 14, 1920, in Mendoza, Argentina, and worked as an engineer before earning a PhD in mathematics.  This early engineering seems to have stuck with him:

[His] revolutionary influence turned the 1950s trend toward abstract mathematics back to the study of mathematics for practical applications in physics, geometry, calculus, and many other branches of this field. His award-winning research in the area of integral operators is an example of his impact on contemporary mathematical analysis. 

(From yourdictionary)

The prizes mentioned above include the Wolf Prize (“for achievements in the interest of mankind and friendly relations among peoples”) and the National Medal of Sciences (for “outstanding contributions to knowledge in the physical, biological, mathematical, or engineering sciences.”)

One of  Calderón’s  graduate students, Carlos E. Kenig, described him in the following way:

Alberto Calderón was a very unassuming man of natural charm, a person of great elegance and restraint, and wonderful company. Mathematically Calderón was exceptional not only for the strength of his talent but for his peculiar way of grasping mathematics. He redid whole theories by himself, got to the core of what he wanted to know by himself, found always his own way. His ideas and the methods he developed were always extremely original and powerful.

(From the AMS Notices)

Alberto Pedro Calderón passed away on April 16, 1998, in Chicago, Illinois.  A biography at the University of Chicago noted:

Calderón is survived by his wife, noted mathematician Alexandra Bellow (née Bagdasar), recently retired from Northwestern University, whom he married in 1989; and two children from his first marriage, Mary Josephine, of St. Charles, Ill., and Pablo, of New York, N.Y.His first wife, Mabel (née Molinelli Wells), to whom he was married for 35 years, died in 1985.

From the biography

Don’t want to wait a whole other week before reading about another mathematician?  The website Lathisms (Latinx and Hispanics in the Mathematical Sciences) is posting a biography every day during Hispanic Heritage Month (Sept 15-Oct 15)

Now on sale!

December 7, 2018 by

It’s the end of the semester, and I’m pretty much out of markers.  So it’s time to order up some new ones.  And hooray, they are on sale!

 

Well sort of.

 

 

 

Or maybe I’ll wait until the sale ends….

 

A Math Mistake in the News, decimal version

August 7, 2018 by

Last month there was a story on BBC.com entitled “Spain’s new submarine ‘too big for its dock'” (https://www.bbc.com/news/world-europe-44871788)

The main part of the story is that Spain’s new non-nuclear submarines were built too large for their docks.  (Hmmm.  Guess that was obvious just from the headline.)

The reason the submarines were too large is that they were redesigned to be bigger than originally planned.

The reason they were designed to be bigger than originally planned is that they were heavier than expected, and so the buoyancy was off, which for submarines is pretty important.  By the time that was discovered it was easier to increase the buoyancy by increasing the volume than by decreasing the weight.

And finally:  The reason that they were so heavy is that someone put a decimal point in the wrong place. According to the article “Navantia gets US help to fix overweight sub” by T. Kington (from http://www.defensenews.com in June 2013, but apparently unavailable now), the former director of the Office of Strategic Assessment at Spain’s Defense Ministry, said “I have been told it was a simple matter of someone writing in one zero when they should have written three.”  I put that in bold, because that small mistake, just twice zero, has taken years and millions billions of euros to (still not fully) rectify.  The contracts for four subs were signed in 2004, the first of the subs was nearly done in 2012 when the mistake was discovered, and now it looks like the subs are all dressed up with nowhere to go.  Poor subs – we look forward to a mathematically successful end to this story.

 

The submarine photo isn’t actually an S-80:  it’s a public domain photo of the USS Chicago (U.S. Navy Photo by Photographer’s Mate 1st Class Kevin H. Tierney. Edited by ed g2s). 

The Sierpinski Cake

June 14, 2018 by

The verdict is in:  despite my less-than-perfect cutting and decorating skills, a Sierpinski cake does indeed look pretty cool.

Sierpinski Triangle made of cake

I had been so distracted by the descriptions of triangular cupcake liners that I didn’t order any, but certainly those would have made for more equally shaped triangles and a crisper looking design.  This was made by baking a 9″×13″ cake, cutting it into 3 long strips, and doing a zigzag pattern on each to get 9 equilateralish triangles plus a bit extra.  Now I just need an extra-large platter for serving, because we are SO having this again sometime…

 

The mysterious triangle

June 12, 2018 by

I was just looking up triangular cupcake molds, as one does — or at least, as one does after the upcoming birthday child asks if their cake can be made into a Sierpinski triangle, and you both mull about it and then realize that that would be CRAZY because then no one would get any cake, and then you get distracted wondering if it could be approximated with cupcakes and you realize that would be awesome and the next time you make cupcakes they really should be displayed in a Sierpinski triangle — and  upon looking at a couple pages of molds I noticed a funny thing.

Amazon has trouble with triangles.

Granted, I only saw oddities on two pages, but since that’s 100% of the pages I looked at, it was still a pretty high percentage.

Exhibit #1:  Triangle is now a color:

Color:Triangle

Exhibit #2:  Triangles have a diameter:

AmazonTriangleDiameter

Of course, triangles do in fact have a diameter, since there’s a maximal distance between points (which turns out to be the length of the longest side).  So presumably these are 2″ per side, and Amazon gets bonus points for an unnecessary use of Diameter.  I admit, though, that part of me thinks that what they called “diameter” was really the distance from a vertex to the opposite side, because from a visual perspective I think that seems the most like the diameter of a circle.  That would mean that the actual side length was (2/√3)*2″, or about 2.3″.  The other cupcake liners were 2″ to a side at the base and 2.4″ to a side at the top, which means that either number would be consistent even if these were the same size.

So in the end, I’m not sure of the answer.  What I am certain of, however, is that the Sierpinski approximation cupcake is an awesome idea, and should become a reality as soon as possible.

 

The rate of change of gasoline

May 4, 2018 by

Is it better to fill up a gas take once a week for $80, or put in a quarter tank 4 times a week for $20 each time?  That question does have two reasonable answers, depending in no small part on whether you have access to $80 or just to $20 at a time, but what isn’t in doubt is that four quarter-fill-ups at $20 each isn’t actually cheaper overall than one full-fill-up at $80.  Or at least, that shouldn’t be in doubt.

There’s an article about it here, but it doesn’t lessen the confusion at all.

(The reference to the question of  which is closer, the West Cost or the Moon, is a reference to a discussion from a year ago.)