Monday Morning Math: Voronoi Diagrams and Georg Voronoy

Good morning!  Today’s math is Voronoi Diagrams, but I’ll lead into it with some Geometry.  Suppose you have two points A and B, and wonder which points on the plane are closer to A and which are closer to B.  Maybe A and B are schools, and this is for figuring out districts, or maybe they are pizza places and you’re wondering where to order from.  It turns out that the border between the two regions is on the perpendicular bisector of A and B.

But what if there are three points? It’s a little more complicated, but the boundaries are still made up from the perpendicular bisectors of the different pairs of points.  And this idea continues even when there are more points, as this picture shows.

Creative Commons by  Balu Ertl

Diagrams like these are called Voronoi Diagrams named after Ukrainian mathematician Georgy Feodosevich Voronoy (also written as Georgii Voronoi).

Wikimedia Commons

He was born in the village of Zhuravka in the north central part of Ukraine on April 28, 1868, and while he was still in the equivalent of high school he solved and published the result of a problem in algebra.  He then went to the University of St. Petersburg in Russia, first as an undergraduate but eventually as a doctoral student under Andrey Markov (himself well known because of something called Markov Chains, which are ways of calculating probabilities).   Both his Master’s thesis and his doctoral thesis were awarded the Bunyakovsky prize for outstanding work in mathematics by the St Petersburg Academy of Sciences.

Voronoy became a professor at the University of Warsaw in Poland, where he continued to do research and also supervise students: one of his students was Wacław Sierpiński, for whom Sierpinski triangles are named.   When he was only 40 years old he developed severe gallstones, and passed away on November 20, 1908.

Sources:

• Wikipedia
• St Andrews
• Mathematical Genealogy
• encyclopedia.com

Monday Morning Math: Presidents’ Day edition

Today is the day we observe Washington’s Birthday, popularly known as Presidents’ Day, and that makes it a good day to look at the Pythagorean Theorem.  Specifically, a proof of the Pythagorean Theorem.  Specifically, the proof created by James A. Garfield, the 20th president of the United States.

The Pythagorean Theorem, often written as , says that squares on the legs of a right triangle, added together, have the same area as a square on the hypotenuse,   Garfield’s proof, published 5 years before he assumed the presidency, used a trapezoid.  Here’s a photo from the New-England Journal of Education on April 1, 1876

(from Mathematical Treasures)

Garfield’s proof compares the area of the trapezoid (the height times the average of the parallel sides) with the area of the three triangles that make up the trapezoid.  Garfield wrote out the details for you above, but feel free to try it using the common a b and c notation.

Monday Morning Math: Valentine’s Day

Happy Valentine’s Day everyone!  I thought I’d give you a valentine.  Maybe a cardioid like this:

By AtomicShoelace – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=39636709

Or, if you don’t want to have to turn your head sideways, here’s a heart made out of Sierpinski Triangles:

Here’s how to make it

And here are interlocking Mobius Valentines!

And here’s how to make these!

Have a happy day!

Monday Morning Math: Gloria Ford Gilmer

Our mathematician today is Dr. Gloria Ford Gilmer, a pioneer in ethnomathematics (the study of the relationships between mathematics and culture).  Gloria Ford Gilmer was born in Baltimore Maryland, in 1928.  She studied mathematics at Morgan State University in Baltimore, where she was a student of Clarence Stephens, and where she earned her bachelor’s degree in 1949.  Two years later she earned her master’s degree in math at the University of Pennsylvania.

She did ballistics research for the US Army, but soon turned to teaching.  She taught both high school and college students, eventually earning a PhD in curriculum and instruction at Marquette University.  Most of Dr. Gilmer’s research was in ethnomathematics.  She was particularly interested in finding mathematics in everyday places, and is known for her mathematical analyses of the braiding patterns in African American women’s hair. 

Dr. Gilmer was active in many professional organizations, and was a “first” for many of them. She was the first African American woman on the board of governors for the Mathematical Association of America, and the first woman to give the Cox-Talbot Address for the National Association of Mathematicians. In 1985 she, along with Ubiratan D’Ambrosio, Gil Cuevas and Rick Scott, co-founded the International Study Group on Ethnomathematics (ISGEm); she served as the organization’s president for 11 years. 

Dr. Gilmer passed away only a few months ago, on August 25, 2021.  The recently established American Mathematical Society’s Claytor-Gilmer Fellowship is named in her honor.

No picture here because I couldn’t find one without copyright restrictions, but you can see one on the site Mathematically Gifted & Black, where she was an honoree last year.  Every day in the month of February the site recognizes a mathematician – check out the 2022 honorees!

Sources:

• Gloria F. GIlmer, a 2021 Honoree for Mathematically Gifted & Black
• Women Who Count: Honoring African American Women Mathematicians by Shelly M. Jones
• “Gloria Ford Gilmer, Ph.D.” obituary in the Milwaukee journal sentinel
• “Gloria Ford Gilmer” in Wikipedia
• “Milwaukee mathematician Dr. Gloria F. Gilmer used creativity, culture to help students succeed” on WTMJ-TV Milwaukee
• “Remembering Gloria Ford Gilmer (1928-2021)” in the AMS (published Oct 21, 2021, but no direct link available: search for Gilmer)