Interesting Geometry from Futility Closet

 Of points, planes, and spindles:

https://www.futilitycloset.com/2022/03/10/the-seven-spot-problem/

Matchstick Beauty

 Once again, an offering from Futility Closet:

https://www.futilitycloset.com/2021/12/16/network/

Albert E. Gives An Assist To a 14-Year-Old

Getting homework help from the best:

https://mindyourdecisions.com/blog/2021/10/26/when-albert-einstein-helped-a-14-year-old-student-with-math-homework/

A Stripey Triangle

 From Presh Talwalkar yet another triangle geometry problem to end the week:

https://mindyourdecisions.com/blog/2021/08/16/triangle-stripes-problem/

More Geometry Whilst Re-visiting John Conway

 A little John Conway courtesy of Futility Closet:

https://www.futilitycloset.com/2021/08/15/the-conway-circle-theorem/

For Geometry and Triangle Fans (especially)

 A fun li'l Twitter geometry thread from yesterday:

https://twitter.com/DavidKButlerUoA/status/1426286914076692480

More Jordan....

 In case you were wanting almost 3 more delicious hours of Jordan Ellenberg hitting on a range of topics (in conversation with Lex Fridman), you got it:

https://www.youtube.com/watch?v=tueAcSiiqYA

Catriona Agg via Aperiodical

 A podcast from Aperiodical with Catriona Agg, geometry-puzzler-extraoridnaire:

https://aperiodical.com/2021/05/mathematical-objects-arbelos/

Manifold Primer

 Nice introduction to manifolds (~50 mins.):

Shearer Delight....

                        
Most readers here likely already know of Catriona Shearer's incredible geometry problems regularly presented on Twitter (though they travel around the Web elsewhere), but if you don't, time to get started with this recent fun one:

https://twitter.com/prdoyle/status/1281905636914204682

A Favorite Space?

Barnes & Noble always has a number of quirky (relatively-inexpensive) 'popular’-style math books scattered around the store giving introductory samplings of a variety of math topics — usually the books are from Britain, often from publishers I’m not familiar with, and I often don’t find them very appealing, but occasionally do.

One I picked up recently is called “Math Hacks” by Rich Cochrane, which, once you get past the first couple dozen topics (out of 100 total), touches on some slightly more advanced topics than is often the case.  I’m enjoying the range/variety of topics mentioned. On the downside, only 1-2 very pithy pages is devoted to each subject, so if you already know the topic, you won’t gain much (if anything at all), and if you don’t already know the topic, you’re not really given enough to grasp it well, despite the over-hyped pitches made for the volume. So I don’t really recommend it other than as a source to dabble with, that might prick one's further interest in some given area.

At any rate, today I ran across “the illumination problem” in it (#73 of the 100), something I’ve mentioned here in the distant past and had forgotten about as an interesting and non-abstract geometric conundrum — it deals with configuring a room of mirrored (light-reflecting) walls in such a manner that a point light source within the room does NOT fill the entire room with light, but leaves some area(s) in the dark.

When I previously posted about it, it was to mention George Tokarsky’s 26-sided polygonal room solution to the problem in 1995, which led in turn to D. Castro's similar 24-sided solution below... a  space that I could imagine Evelyn Lamb enjoying ;)

via HERE

(If light source is at point "A" then point "B," amazingly, is in the dark.)

There are other solutions (not all polygonal) to the problem, and the Wikipedia take on it is here:

https://en.wikipedia.org/wiki/Illumination_problem

…meanwhile the Numberphile treatment here:

Ford Circles

(via WikimediaCommons)

Awhile back I mentioned Alfred Posamentier’s latest volume “The Circle,” and around now it should be showing up in bookstores -- another great little geometry offering from Dr. Posamentier (and Robert Geretschlager). One of so many interesting tidbits in it is about “Ford circles”:

Imagine you have two tangent circles sitting atop a number line, one tangent to that line at “0” and the other tangent at “1.” Now in the space between these circles draw another circle tangent to both the “parent” circles and to the number line as well — it will touch the number line at the 1/2 position. You can keep iteratively drawing such circles (to infinity) in the space created with each new (smaller) circle. Now, quoting from the book:

“Of course, the circles get very small very quickly. As it turns out points of tangency of all these infinitely many circles with the number line have a quite unexpected property. The points of tangency are precisely the rational numbers in the interval between 0 and 1. No circle created by this process touches the number line at an irrational point, and every rational number is the point of tangency for some circle created in this manner.”

Pretty amazing, and a nice demonstration of one area of mathematics, plane geometry, connecting to other areas of infinity and number theory. Further, these circles relate back to Farey sequences.

Here’s one of several treatments of Ford circles on the Web:

https://nrich.maths.org/6594

Old, Old Favorite

Below, a problem I gave to a local math meetup group some weeks back thinking it would be familiar to EVERYone (it is my favorite problem from geometry class that I had 50 years ago(!), and I've seen it on the Web multiple times). To my surprise, NO ONE was familiar with it, so I present it here in case anyone else has led such a sheltered life as to have missed this delightful puzzle ;-):

MNOP is a RECTANGLE inscribed in one quadrant of a circle ("O" being the origin/center of the circle).
PO = 10 cm.
SP= 3 cm.

What is the length of diagonal line PN?
(solution can be arrived at in seconds with no trig and very little geometry required)
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answer below:
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ANSWER:  as a diagonal, PN is EQUAL to MO (not drawn in). MO is a radius, as is SO. SO=13, thus PN=13.

Celebrating the Season With a Tribute and Some Geometry

Pat Ballew 'celebrates the season' this morning with some "beautiful geometry" from "a little known mathematical dilettante":

http://pballew.blogspot.com/2015/12/the-geometric-beauty-of-troubled-mind.html

Pat writes that George Odom Jr. "found five different simple geometrical approaches to the golden ratio using equilateral triangles, and platonic solids" that "are too beautiful to be so unknown." A nice tribute to someone likely unknown to most of us.

Also, a wonderful, 2007 piece by Siobhan Roberts (...you may have heard of her) on Odom, and his connection to John Conway, here:
http://thewalrus.ca/2007-04-field-notes-2/

Tiling the Plane... Again!

Well, this is sort of neat news: discovery of a new (15th) convex pentagon that successfully tiles the plane:

https://plus.google.com/u/0/100003628603413742554/posts/JC8hqyUHuDE

 If you don't know the history of this problem you can check out the Wikipedia article that is linked to in the above piece. Following additional findings after Martin Gardner originally drew attention to the problem back in the 1970s, the number of such tiling forms had stood at 14 for 30 years!
How many more are there???

Elliptical Pool!

via ClipArt etc.

So far as I know, the game of pool hasn't changed much in a long time... until NOW! Alex Bellos has created a fun, new version of pool (he calls "LOOP") based on an ellipse. How cool is that! Take a look:

http://tinyurl.com/p8g5236

Geometry Fun

Just a few past Futility Closet geometry puzzles to get your week rolling:

http://www.futilitycloset.com/2014/11/25/sizing-up/

http://www.futilitycloset.com/2014/10/13/stretch-goals/

http://www.futilitycloset.com/2012/07/21/tall-and-wide/

A Congruency of Triangles

Futility Closet reports succinctly on a recent geometric theorem from the ever-interesting Lee Sallows:

http://www.futilitycloset.com/2015/02/19/triplets/

Seems odd that this simple finding on triangles hasn't been reported previously elsewhere(???), but that's part of the beauty of mathematics, that such elementary conclusions can be sitting out there just waiting to be discovered.
Now, does this one have any particular applications...?

Math, Women, Tessellation, Intuition

(image: WikimediaCommons)

A lot of discussion around the Web these days about women in STEM, and at Math-Frolic I'm even more interested in women in math, so thought it would be fun/timely to recount the unusual story of Marjorie Rice -- worth repeating, even if most are familiar with it, as a rare instance of someone becoming involved with math almost by accident.
[Most of this information was reported over a year ago in a MathMunch piece on Marjorie here:
http://mathmunch.org/2013/02/25/marjorie-rice-inspired-by-math-and-subways/  also see Ivars Peterson's 2010 piece here: http://mathtourist.blogspot.com/2010/06/tiling-with-pentagons.html ]

Marjorie discovered her senior year in high school that she found math interesting, but by then it was too late to do much with it. She went on to marry, have children, be a housewife; i.e. she took NO mathematics past high school. But after getting a subscription to Scientific American for her son, she began reading the Mathematical Games column of Martin Gardner, including a 1975 column concerning "pentagon tessellations," i.e. pentagon forms that could cover an entire plane, repeating themselves with no gaps, like a jigsaw puzzle. At one time mathematicians believed there were only five such pentagon shapes that achieved tessellation, but in 1968 three more were discovered, and a fourth new one had just been added in 1975 that Gardner was reporting on.

Marjorie was intrigued. And playing with different pentagons, with different internal angles, she finally found a fresh one that accomplished the feat of tessellation. Inventing her own unconventional notation to describe her work she wrote to Gardner showing the result. And he sent her correspondence on to another female mathematician, Doris Schattschneider, who confirmed Marjorie's success and translated her work into more standard mathematical format.  Marjorie went on to find yet three more successful pentagon tessellations, and also DISproved a conjecture made by Doris.
Successful amateurs have made significant contributions to astronomy, but in most sciences, and particularly in mathematics, it is rare for an academically-untrained amateur to accomplish something missed by professionals... but apparently Marjorie didn't know that! Her own website on her work is here:
https://sites.google.com/site/intriguingtessellations/home

She is now over ninety, and remains an inspiration, not just to women, but to amateur enthusiasts everywhere. What I love most though about the Marjorie Rice story isn't that she was a female in mathematics, nor even that she was an amateur contributing to a technical field, but rather what the story says about the role of intuition and insight in math. Despite mathematics' image of being cold, dry, and rigid, and despite its abstractness in advanced study, scrape below the surface and there remains, on occasion, a powerful substrate of intuition and mental imagery, accessible to many.

Below is a video segment (from about the 31:50 point to 35:45) talking about Marjorie's work (again h/t to MathMunch for this):





We now know of 14 tessellating pentagon forms! Are there more?

Geometry and Ancient Dance

George Hart (perhaps better known these days as Vi Hart's father ;-)) brings us an introduction to the mathematical aspects of the ancient art form of "long sword dancing":

http://tinyurl.com/kn8ckat

Combines dance/movement, geometry, structure, and sheer gee-whizness! Fun to watch.