Monday Morning Math: Illusions

Good morning!  Today’s post is about illusions – specifically, the kind of illusions you can verify with MATH!  

I kind of think calling them “Illusions” might be giving some answers away, but even suspecting the answers it can take some convincing, possibly by holding up paper to your phone or computer.

Which center circle is bigger?

Would lines m and n meet if they were extended?

Which segment is longer, AB or CD?

Will line m, when extended, meet point A, point B, point C, or none of these points?

We’ve posted about illusions before (e.g. here and here) and if you want to see more you can go to the contest for the Illusions of the Year! They include photos and videos of the best ones, with some hints of how they work:

Enjoy!

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: Dorothy Lewis Bernstein

Happy birthday Dorothy!  Today is the 108th birthday of the first woman to be president of the Mathematical Association of America.

Dorothy Lewis Bernstein was born on April 11, 1916, in Chicago, Illinois.  Her mother, Tillie Loyev (changed to Lewis upon immigration to the US), was born in Ukraine and her father, Jacob Louis Bernstein, in Russia.  Her parents married in Milwaukee (where her mother’s family lived) in 1912 and then moved to Chicago where her father was a dairy farmer.  Dorothy was the oldest of six children and grew up in Chicago and Wisconsin.  

Dorothy earned her BA and MA in mathematics from the University of Wisconsin Madison [the alma mater of yours truly.  Go Badgers!] and her PhD from Brown University.  She worked at several universities, including Berkeley where work was being done to support the army during World War II.  She then spent 14 years just down the street from Naz at the University of Rochester(!!!).

While at Rochester she was asked by C. B. Tompkins, who was working at Engineering Research Associates on a contract from the Office of Naval Research, to undertake a study of the current state of knowledge of existence theorems in partial differential equations. As she explained in her 1978 AWM talk, “some of the proofs could be used as basis for the computational solutions of non-linear problems that were just being tackled by high-speed digital computers.” Her 1950 book with Princeton University Press was the result of this undertaking. 

Supplementary Material for Pioneering Women in American Mathematics: The Pre-1940 PhDs

In 1959 she began working at Goucher College in Maryland, where she stayed 21 years.  Dr. Bernstein was a prominent researcher, and was active in several organizations both before and after her retirement in 1979.  She was the vice president for the Mathematician Association of America in 1972-73 and president (the first woman elected to that position) 1978-80.  She passed away in 1988 at the age of 73.

Sources:

• MAA
• MacTutor
• Supplementary Material for Pioneering Women in American Mathematics: The Pre-1940 PhDs
• Wikipedia

Monday Morning Math: Angles in a Triangle

Good morning!  I’m teaching Geometry this semester, so triangles are on my mind.  And here are some facts about triangles that, even though I know them, still blow me away.

I was taught that the angles in a triangle add up to 180 degrees.  This is true, at least in Euclidean Geometry, which is geometry done on a flat surface like a plane or piece of paper..(That link was to an illustration using folded paper but there are many proofs too.)

But what if you’re not on a flat surface?  What if you’re on a sphere?  Then things get weird.  “Lines” on a sphere are defined by the shortest path between two points (formally called geodesics), and it turns out that if you draw a line between two points and keep going, it will cut the sphere in half, like the equator of a globe, or a line of longitude.  And if you put three of those together you get a triangle…but the angles don’t add to 180 degrees.  In fact, you can have have a triangle with not one but two right angles, as shown below:

CC by Lars H. Rohwedder, Sarregouset

You can even have a triangle with three right angles!  Or three obtuse angles!  And overall, there isn’t a fixed amount that the angles add up to: it can be anything from just over 180 degrees all the way up to just under 540 degrees (which would be a triangle with three really big angles, covering a significant amount of the sphere).   So weird.

That’s not the only thing that’s weird.  If, instead of being on a sphere (which bows out), you are on a hyperbolic paraboloid, which bows in and looks like a saddle, then everything is opposite and the angles of a triangle are smaller than they would be on a flat surface.  You can still have triangles with one right angle or one obtuse angle, but the other two angles will be a bit smaller, and some triangles will just have three very small angles.  As in spherical geometry, the angles of a triangle don’t add up to any one fixed amount, but can be any positive number less than 180 degrees. 

Public Domain

If you like exploring, the (free) program Geogebra [that mimics drawing with a straightedge and compass]  has tools that let you draw in spherical geometry and hyperbolic geometry. Enjoy!