Monday Morning Math: D. R. Kaprekar and his constant

Happy May Day!  It’s finals week here, and so this will be the last Monday Morning Math until sometime in September. But in the meantime, here’s a neat magic trick

• Start with a 4 digit number. The digits don’t have to all be different, but they can’t all be the same (so 1001 is OK but 1111 is not).
  
• Now look at the 4 digits, and write down both the largest number and the smallest number you can make with those digits: in my example the largest would be 1100 and the smallest 0011.   
  
• Take the difference:  I got 1100-0011, which is 1089.  (Hey, that’s pretty cool!  I’ve seen that number before.  But I digress.)
  
• Now repeat the process: take your new 4 digit number, write the largest and the smallest number you can make with those digits and take the difference.  And keep doing it.  After a while, you’ll get the same number over and over again.  And that number is……

6174

This is called Kaprekar’s constant, named after Dattatreya Ramchandra Kaprekar.  D. R. Kaprekar was born on January 17, 1905, in Dahanu, India, and enjoyed math puzzles and numbers from when he was young.  He studied math in college and in 1927, when he was 22, was awarded the R. P. Paranjpe Mathematical Prize for original work.  His discovery of the pattern above isn’t his only work, but it might be the best suited for magic tricks. 

There’s a  version with 3 digits too,  but once you get to 5 or more digits you start getting cycles instead of the same number over and over.  And that’s just Base Ten – you can explore other bases to see what happens. Enjoy!

Sources:

• Wikipedia
• MacTutor (where the picture came from)
• Hat tip to Ricardo Teixeira, whose presentation on mathemagic introduced me to this constant!

Monday Morning Math: Katherine Johnson

Good morning! This week, on February 24, marks three years since Katherine Johnson passed away, and it seems a good opportunity to write about her.  

Katherine Coleman was born on August 26, 1918, in White Sulphur Springs, West Virginia.  Her mother, Joylette, was a teacher and her father, Joshua, a farmer; she also had three older siblings: Charles, Margaret and Horace.  The school for African Americans in White Sulphur Springs only went through the 8th grade, so the family moved to where the kids could get more schooling.

Katherine’s father had been good at math and Katherine was too.  Very good. Indeed, she was quite good in many subjects. She skipped a few grades, started high school when she was ten, and was supported and encouraged by her family and teachers.  She graduated summa cum laude from West Virginia State College in 1937 with degrees in French and mathematics, and began teaching school herself at the age of 19.   Two years later West Virginia University began to integrate its graduate school, and Katherine attended classes for a time, and then married James Francis Goble.  They had three children: Constance, Joylette and Kathy, and after several years Katherine returned to teaching.  

In 1953 Katherine and her family moved to Newport News, Virginia, so she could work in nearby Hampton at the Langley Research Center, which was part of the National Advisory Committee for Aeronautics (NACA) [which eventually became NASA]. She started as a computer, a person who performs mathematical calculations, but she and a colleague were soon assigned to what was supposed to be a temporary assignment with the (then all male) flight research team, where she worked for several years.

James Goble passed away in 1956 after a several-years battle with cancer.  Three years later Katherine married James A. Johnson, whom she had met through her minister. Throughout all this Katherine continued her work, performing calculations for Alan Shepard’s Mercury mission in 1961, John Glenn’s orbit around the earth in 1966, and the moon landing in 1969.  She continued working at NASA for more than 30 years, and during that time she co-authored a book on space and dozens of research articles, and continued work in many areas such as .  She also worked on the Space Shuttle program and an eventual mission to Mars. 

Katherine Johnson retired in 1986 and in 2015 was awarded the Presidential Medal of Freedom by President Barack Obama. Around the same time the book Hidden Figures by Margot Lee Shetterly was published and became a major film, allowing many people to learn of all that Katherine Johnson had accomplished.  Her own daughters, too, followed in her footsteps: Constance and Kathy became educators, and Joylette a computer analyst at Lockheed Martin.

Katherine Johnson passed away on February 24, 2020, at the age of 101.  In addition to Hidden Figures there are several other biographies of her, including one that she herself wrote for children and young adults: Reaching for the Moon.

Sources:

• Visionaryproject.org
• Thehistorymakers.com
• Nasa.gov
• Space.com
• Biography.com

Monday Morning Math: Happy New Year!

Good morning everyone! Happy 2023!    Today’s post is in honor of two new years.

The first is the new calendar year: 2023.  If you want to do something fun (which of course you do) then you can see if you can use exactly the digits 2, 0, 2 and 3  and different math configurations to write the numbers 1-100.  For example: .  Or  .  (There are examples all over the internet, so a quick search reveals many solutions should you wish.)

The other is the Lunar New Year, which began yesterday.  This is also known as the Spring Festival, and is observed around the world, including China, Indonesia, Japan, Malaysia, the Philippines, Singapore, South Korea, Taiwan, Thailand, and Vietnam: we are now in the Year of the Water Rabbit in many countries, and the Year of the Cat in Vietnam.

In honor of the New Year we’ll talk about the mathematician Jing Fang  京房.  He was born in China 2100 years ago (78 BCE, during the Han Dynasty).  He was a mathematician and, appropriately enough for this post, he described astronomy – the solar and lunar eclipses.

But these weren’t his only accomplishments.  He was also very good at making predictions using the Yijing, or I Ching. There are 64 hexagrams, each made up of 6 rows that have one long or two short marks in each row.  While this isn’t mathematics, it does lead to the math question to ponder: namely, can you explain why there are exactly 64 configurations?

Picture from Wikimedia showing a diagram that Gottfried Wilhelm Leibniz owned.

And his math-connection doesn’t end there either – he used mathematics to describe music theory, particularly that 53 fifths (technically “just fifths”, which may or may not be the same as a perfect fifth – but you can here one here) was almost exactly 31 octaves.  It took more than 1600 years for anyone (said anyone being Nicholas Mercator) to caculate the difference between the two more exactly than Jing Fang. 

Math Moons and Music – a good way to start the year.

Sources: Wikipedia and…just that, because the few other sources I found had the same information.

Monday Morning Math: Leonardo Pisano (Fibonacci)

Good morning!  I hope you all enjoyed your Thanksgiving Holidays. Both boys came home for the weekend, which was a treat. =)

On Wednesday I got a text wishing me a Happy Fibonacci Day.  I had to think a minute: Nov 23, so 1123, from the sequence 1, 1, 2, 3, 5, 8, 13,….  And this inspired this week’s post about Fibonacci.  

Leonardo Pisano was born around 1170 in Italy, probably Pisa — hence the “Pisano” part of his name.  He was born to the Bonacci family — hence the  “filuis Bonacci” (abbreviated to “Fibonacci”) part of his name.  His father was a diplomat, and as a result of his father’s post Leonardo was educated in North Africa and traveled widely, which meant he was exposed to different number systems, including the base ten number system that we use today.  Indeed, it is likely that Leonardo himself is the reason we use it: he found it to be much better for calculation than the Roman number system (which would have used XXIII for a number like 23).  He returned to Pisa around 1200 and wrote several books that illustrated this system, the most famous of which is Liber Abaci (Book of Calculation – abaci is related to abacus).  Here’s a statue, by Giovanni Paganucci, of Fibonacci holding a book (CC license).

Although Fibonacci’s most significant mathematical contributions are related to his books sharing the decimal number system and methods of calculation with western Europe, he has become most famous because of a single problem that was in the book:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

This problem leads to the number system 1, 1, 2, 3, 5, 8, …, where each number is the sum of the previous two, which now bears the name the Fibonacci sequence in his honor.  Although maybe it shouldn’t – the sequence was known in India well before Fibonacci. By whom, you might wonder?  I started to write a brief summary, but realized I didn’t know enough about the history myself to do it justice so that will have to wait for next week…

Sources: 

• http://www-groups.dcs.st-and.ac.uk/history/Biographies/Fibonacci.html
• “The so-called fibonacci numbers in ancient and medieval India” by ParmanandSingh, Historia Mathematica, Vol. 12, Issue 3, August 1985, pp. 229-244 https://www.sciencedirect.com/science/article/pii/0315086085900217?via%3Dihub

Monday Morning Math: Omar Khayyam

Good morning!  Our mathematician today is Omar Khayyam.

CC license; not attributed

Omar/Umar Khayyam was born in Nishapur, Persia, (modern day Iran) in 1048. Not much is known about his mother, but his father was a doctor who hired tutors to teach Omar.   Omar Khayyam is known for his mathematics, including writing down the laws of algebra that we know today.  He was able to make progress toward finding a general formula for  similar to the quadratic formula:  Greek mathematicians had come up with solutions to the quadratic formula that used a straightedge and compass, but Khayyam conjectured that it was not possible to solve the cubic equation with just those tools, and so developed other means of finding the solutions geometrically, using a parabola.  (It would be more accurate to say solutions to cubic equations: although we write it as a single equation, at that time the quadratic and cubic equations were written as several different cases depending on whether the coefficients were positive or negative.)  It was 500 years before anyone found a more general solution than his.

Omar Khayyam was one of the earliest people to describe the Arithmetic triangle (which is sometimes called Pascal’s triangle, although this was 500 years before Blaise Pascal).  He also contributed to the fields of non-Euclidean geometry and number theory.

In addition to mathematics, Khayyam wrote about astronomy, geography, and music.  He is largely remembered for his poetry, especially the rubaiyat (aka  Rubā‘iyyāt, or quatrains)

The Moving Finger writes, and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.

(Translated into English by Edward Fitzgerald)

Khayamm passed away in Nishapur in 1131, and a mausoleum stands over his tomb

Sources:

• www.famousscientists.org/omar-khayyam/
• www-groups.dcs.st-and.ac.uk/history/Biographies/Khayyam.html
• plato.stanford.edu/entries/umar-khayyam/
• en.wikipedia.org/wiki/Omar_Khayyam

Monday Morning Math: The L’Hôpitals

Good morning! The math tidbit for today is a two-for-one special: the L’Hôpitals, who lived in France in the 1600s.  But we’ll start with the phrase that my brain jumps to when I see L’Hôpital, which is L’Hôpital’s Rule (also called L’Hospital’s Rule).  It’s about limits, so it shows up in Calculus.  Essentially, if you have a limit as of a fraction where both the numerator and denominator individually are approaching 0 or where both the numerator and denominator individually are approaching , then:

provided that second limit actually exists.  So, for example,

.

Huzzah!

This rule was named after Guillaume François Antoine de l’Hôpital, where that last name was spelled different ways even in his lifetime, even by him.  He learned calculus by correspondence with Johann Bernoulli, who was busy figuring out calculus himself at that time, since the subject was only a few decades old.  In 1696 l’Hôpital published what is considered to be the first Calculus book: Analyse des infiniment petits pour l’intelligence des lignes courbes.  He thanked several people in the introduction, including Johann Bernoulli, but it wasn’t clear at that time how much of the book was really after l’Hôpital’s own work (none?) and how much was based on Bernoulli’s notes (all?).  Apparently Bernoulli was fine with l’Hôpital publishing the book, possibly because of the money l’Hôpital paid him, possibly because he was happy just to have these still-new ideas disseminated. L’Hôpital died in 1704 when he was about 43 years old.

A lot of that information  comes from the MacTutor biography, which also states, “L’Hôpital married Marie-Charlotte de Romilley de La Chesnelaye; they had one son and three daughters.”   The English Wikipedia page adds that his wife was “also a mathematician and a member of the nobility, and inheritor of large estates in Brittany” with a link to a page for her – in French – from a French biography that indicates that she worked in Geometry and Algebra and lived from 1671-1737.  The only other site online that mentions her is this dictionary, also in French, which indicates that she helped with the printing of the aforementioned Calculus book and impressed another math professor, Monsieur de la Montre, with her knowledge of Euclid.  So was she too involved in the creation of that first Calculus book but not mentioned?  It sounds like it, though the extent of her involvement is unclear.  

And we’ll leave on that uncertain note.  Any new information would of course be welcome.

References:

• Wikipedia (English and French)
• MacTutor
• this dictionary
• some pictures of the book

Monday Morning Math: Maryam Mirzakhani

It’s Women’s History Month, and we’ll celebrate with the first (and so far only) woman to receive the Fields Medal in Mathematics – an award given every four years in recognition of outstanding mathematics. 

Photo by Gert-Martin Greuel, Creative Commons

Maryam Mirzakhani was born in Tehran, Iran, in 1977.  She loved reading novels and writing as a child, and became interested in mathematics when her brother would come home from school and tell her what he learned – her earliest memory of math specifically was when he told her the story of a young Carl Friedrich Gauss being asked to add the numbers from 1 to 100 and surprising his teacher by seeing a pattern that allowed him to do the computation in moments.

Maryam Mirzakhani went on to participate in (and win) the Math Olympiads as a teen; she then went to Sharif University of Technology in Iran and then Harvard University in the United States, where she earned her doctorate.  

In 2014 Dr. Mirzakhani was awarded the Fields Medal. Her research is sometimes described as a complicated version of billiards, where a ball bounces along the edges of a table.  “You want to see the trajectory of the ball,” Dr. Mirzakhani explained in a video produced by the Simons Foundation and the International Mathematical Union. “Would it cover all your billiard table? Can you find closed billiards paths? And interestingly enough, this is an open question in general.” You can see a short (under 3 minute) video of Dr. Mirzakhani explaining her work here:

Mirzakahni passed away in 2017 from breast cancer.  She was 40 years old.

The 2022 Fields Medals will be given out at the  International Congress of Mathematicians this summer.  The conference was originally scheduled to be held in St. Petersburg, Russia, but due to the Russian invasion of Ukraine the organizers have announced that the conference will be held virtually, without any contributions from the Russian Government, and will be free for everyone who would like to attend.  

Sources:

• The Guardian
• Scientific American
• Stanford News

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: 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)

Monday Morning Math: Pythagoras

This week’s mathematician is someone you have probably heard about, although it turns out very little is known for certain. 

The mathematician is Pythagoras.  

From Thomas Stanley, (1655), The history of philosophy: containing the lives, opinions, actions and Discourses of the Philosophers of every Sect, illustrated with effigies of divers of them.

Pythagoras was probably born in Greece, on the island of Samos, over 2500 years ago (570 BCE, plus or minus a few decades).  His mother was from that island, and his father Mnesarchus was a merchant.  Pythagoras would travel with him sometimes when he was a child, and when he was an adult he studied mathematics with Thales, another now-famous mathematician.   

After many years (decades) of study, Pythagoras formed a group known as the Pythagoreans, who followed a strict vegan diet and believed that everything was essentially a number.  There was an inner circle of mathematicians and an outer circle, and there is some indication that the groups were equally welcoming to women and men. It is not possible to distinguish who proved any one result (because secrecy was the name of the game rather than publishing), but there were important results from this group related to music and geometry.  

Two of the most significant results attributed to Pythagoras are the Pythagorean Theorem (written in geometric terms that a square on the hypotenuse of a right triangle has the same area as the sum of squares on each of the two legs) and that the square root of 2 is irrational (written in geometric terms that the side and the diagonal of a square are incommensurable, meaning there’s no teeny tiny amount that fits into both an integer number of times.  There are stories that someone figured this out and was killed, either for figuring this out or for telling people outside the Pythagoreans, but like the rest of what I’ve written this was a story – everything we supposedly know about Pythagoras comes from reports well after his death, so at best this is educated guesswork.

Etching on the wall of Peckham Hall, our math and science building. Not seen is the QED in the lower right corner, as it was also in darkness when I took the photo just now

Finally, the video below, from BBC learning, is under 5 minutes and fun to watch, though it does start off with an excited “Pythagoras!” that will get the attention of anyone around you.

Other sources:

• https://www.famousscientists.org/pythagoras/
• https://mathshistory.st-andrews.ac.uk/Biographies/Pythagoras/

Monday Morning Math: Dorothy Vaughn

PC: https://www.nasa.gov/content/dorothy-vaughan-biography

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.

PC: https://scientificwomen.net/women/vaughan-dorothy-103

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: Grace Murray Hopper

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

US Navy photo, public domain

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:

• An interview with David Letterman
• www.computersciencelab.com/ComputerHistory/HistoryPt3.htm
• www.cs.yale.edu/homes/tap/Files/hopper-story.html
• www.biography.com/people/grace-hopper-21406809
• www-groups.dcs.st-and.ac.uk/history/Biographies/Hopper.html

Monday Morning Math: Adolphe Quetelet

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

Photo credit: https://mathshistory.st-andrews.ac.uk/Biographies/Quetelet/pictdisplay/

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.

Photo credit: https://www.famousscientists.org/adolphe-quetelet/

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

Math Guys in Rome

The Villa Borghese Gardens form a giant park in Rome, and at the western edge of it are the Pincian Gardens, so named because they’re at the top of the Pincian Hill.  (Belated note to self: the fact that they were on top of a hill means it should not have been any sort of surprise that there were many many steps to get up to the Gardens.)

These were [this was?] the first public park opened in the city, and around 1850 a bunch of busts of prominent Italians were commissioned for the park.  Some of these were kept in the park, some were moved and then moved again, and some were altered to represent Italians who seemed more worthy of being commemorated.   Then through the 1950s more busts were added and there are now a total of 228, of which 225 are of men and 3 are of women.

There’s a map of all the busts online [here, along with all the history], so it was pretty easy to search out mathematicians.  Here’s Archimedes:

This was one of the original busts, but back then it was of Niccolò Machiavell; it got re-formed into Archimedes around 1860. (You might be wondering, too, at Archimedes Italian background.    A few of these busts were a little more liberal than others on what it meant to be Italian.)

This next one is of Giordano Bruno, born in 1548:

Bruno was a big fan of Copernicus’s still-unpopular view that the earth revolves around the sun, though he also thought that the sun was nothing unique either — just one of an infinite number of heavenly bodies.  Poor Bruno didn’t get along too well with the church of the time, and was burned at the stake in 1600.

On a lighter note, here’s Leonardo da Vinci, along with a rose that someone left for him:

(I just noticed the square around his face.  What’s that about?  It’s in the few other pictures that we took of him, too.)

Next up is Giuseppe Luigi Lagrangia, also known as Joseph-Louis Lagrange (though Wikipedia and Mactutor say his middle name was Lodovico originally).

He looks totally proud of everything named after him, like the Lagrangian and Lagrange Multipliers.

Here’s Pythagoras (another “Italian”) with two of his closest friends:

And finally, this is Niccolò Fontana, who became known as Tartaglia (stutterer) because the French invaded his hometown of Brescia when he was a teen and sliced his face. Ugh.

He translated Euclid into Italian and is also known for his role is finding a general solution to the cubic equation, which deserves a post all to itself someday.

And that’s it!  In theory Galileo should be in this group, but we couldn’t find him (we think he was hidden behind a construction fence), and so should Barnaba Tortolini (not sure why we missed him).  Oh, and there was also an obelisk and this really cool water clock, which was one of the main reasons that we went to this neck of the woods in the first place, but that will appear in the next post…

RIP Martin

Martin Gardner passed away yesterday (May 22) at the age of 95 — a number that is 0 mod 1, 1 mod 2, 2 mod 3, and 3 mod 4, as I’m sure he’d appreciate.    It seems like half the puzzles I hear about were either invented by him or popularized by him.  Falling into the latter category are the flexagons, which are, of course, a favorite of Godzilla and which were described in the first “Mathematical Games” column of Scientific American (you know, according to Wikipedia).

A quick search reveals lots of puzzles, most of which have a visual component.  As a treat, here’s a quick one that doesn’t (from the group Gathering for Gardner):

• Write out the alphabet starting at J (and ending at I):
  JKLMNOPQRSTUVWXYZABCDEFGHI

  Now erase all the letters that have vertical symmetry, like M.

  There will be 5 groups of consecutive letters left.  Write the number of letters in each group:

  ___   ___   ___   ___   ___

Cool, huh?

Scientific American itself just posted 3 Gardner puzzles here in tribute to the man who served them well for many years.

Martin, thanks for the fun.

The photo of Martin Gardner was taken by Konrad Jacobs of  Erlangen and is licensed under CC.  From Wikipedia.