The Geometry of Gerrymandering

gerrymandering: the practice of dividing or arranging 
a territorial unit into election districts in a way 
that gives one political party an unfair advantage in elections

       A recent Scientific American article by Manon Bischoff, "Geometry Reveals the Tricks Behind Gerrymandering," has reminded me of the horrors of this practice.  To express my thoughts about a particular concept, often a stanza that matches mathematical constraints helps me to carefully consider word choices and attempt clear and concise expression. The following syllable-square is a start toward expressing my point of view:

          For fair elections
          voting districts must
          be proportional,
          not maneuvered by
          gerrymandering.

This Scientific American author Manon Bischoff is an editor at Spektrum der Wissenschaft. She primarily covers mathematics and computer science and writes the column The Fabulous World of Mathematics. Bischoff studied physics at Technical University of Darmstadt in Germany and then worked as a research assistant at Johannes Gutenberg University Mainz in Germany.

Trying a Tritina

      Writer and scholar Marian Christie (born in Zimbabwe and now in Kent, England) has had a long term interest in mathematics and poetry and, during the last several years, she has created a blog -- Poetry and Mathematics -- in which she explores, with careful detail, some interesting and important links between these two arts.

     Christie's work has been featured several times in this blog and my posting today shows my attempt to learn from one of her postings.  At this link, on July 13, 2022, Christie posted "Turning in Circles -- the Tritina" and I have used her posting to learn the requirements for a tritina and, then, to try to write one.

     A tritina consists of ten lines -- three three-line stanzas with a final, separate line.  The stanzas have the same three end-words, rotated in the sequence 123, 312, 231, and a single final line containing all three end-words. 

     I have tried to write a tritina and offer my example below -- not because it is good but because it explores a pattern that I think might work well for students trying to write a poem in a math class.     

ARE THINGS DIFFERENT NOW IN SCHOOL?     a sample tritina     

Student Essay Contest -- Write about a Math-Woman

 Essay Contest -- Sponsored by AWM and Math for America

     Each year the Association for Women in Mathematics (AWM) and Math for America  co-sponsor a contest for essays written about the lives and works of contemporary women mathematicians and statisticians in academic, industrial, and government careers. 

     Each essay should be based primarily on an interview with a woman currently working in or retired from a mathematical sciences career. Participation is open to three groups -- middle school, high school, and undergraduate students.  Submissions open December 1 and continue to February 1, 2023.  Complete submission information may be found at this link.   (AND, 2022 winning essays may be found here.)

     I close with a poem about a math-woman -- "San Antonio, January, 1993" -- a poem inspired by my time at a long-ago mathematics conference and included in a chapbook of my mathy poems, My Dance is Mathematics (available at this link). 

Who is the GOD of ARITHMETIC?

     Recently I have learned (from poet and Capillano University professor Lisa Lajeunesse -- who enjoys linking mathematics and the arts) of the work of Canadian poet Lorna Crozier.  Author of more than a dozen poetry collections and recipient of five honorary degrees, Crozier is versatile and widely read.   Here is one of her fascinating poems:

     God of ARITHMETIC      by Lorna Crozier

     Most children no longer know who this god is. For one thing,
     he uses chalk as if time does everything but erase. In aban-
     doned country schools, he prints columns of numbers on the
     blackboards. There are no pupils to add them up and call
     out the answers though his pockets burn with stars to give
     away. His worshippers, in danger of dying out, recite the
     time tables like Hail Marys under their breath to prove their
     minds are still okay. No matter what they’ve lost—the word
     geranium, the birthdates of their children—they can do their
     sums. He wanted his only commandment to be included on
     the tablets Moses brought down from the mountain, but the
     others, bartering for space, thought it was only about arithme-
     tic and left it out. It would have changed the world. It would
     have made us kinder. Thou shalt carry the one, he intones to
     the small desks in empty classrooms, carry the one.

Copyright © Lorna Crozier. Originally published in God of Shadows (McClelland & Stewart/Random House, 2018). 

One Idea May Hide Another . . .

     One of the excitements I find in both mathematics and poetry is the continuing discovery of new meaning.  A first reading discovers something but subsequent readings discover more and more.  A poem by Kenneth Koch (1925-2002), "One Train May Hide Another," opens with "In a poem, one line may hide another line" -- focusing also on the idea that one thought may obscure another.

     Koch's poem is one that I first met lots of years ago when I was working with middle school students in a poetry class at a newly established Children's Museum in Bloomsburg, Pennsylvania.  At the time, the poem excited me by bringing back memories of traveling through western Pennsylvania as a child when my parents' car often needed to obey flashing red lights and stop while a train crossed our highway.  And sometimes there were parallel sets of tracks and the possibility that two trains might be passing our intersection in opposite directions at the same time. 

     I offer below the opening lines of the poem and a link to the complete poem; I post it with the hope that you also will enjoy it -- and will reflect on the ways that (in mathematics and elsewhere) one idea may hide -- or lead to -- another.   

How we learn Mathematics

      Recently I came across an interesting article about how we learn mathematics by Shaneen Suhail (a Masters student at JK Institute of Mathematical Sciences, Srinagar, India) -- and published online here in Kashmir Reader.   Suhail's thoughtful comments offer many ideas for teachers and students to consider -- and they include comparison of mathematics to poetry!

Like poetry, "mathematics says a lot with a little".  

The Secret Sits     by Robert Frost (1874-1963)

 We dance round in a ring and suppose,
 But the Secret sits in the middle and knows.

Lots more of Frost's words are available here.

Struggling to create -- slave and master . . .

      In the sonnet below, Edwin Arlington Robinson (1869-1935) speaks of the enslavement of a writer of poetry in the effort to explain ideas in a perfect form . . . an enslavement perhaps (or not) also shared by mathematicians.     Food for thought!

       SONNET     by Edward Arlington Robinson

       The master and the slave go hand in hand,
       Though touch be lost.  The poet is a slave,
       And there be kings do sorrowfully crave
       The joyance that a scullion may command.
       But, ah, the sonnet-slave must understand
       The mission of his bondage, or the grave
       May clasp his bones, or ever he shall save,
       The perfect word that is the poet's wand.

       The sonnet is a crown, whereof the rhymes
       Are for Thought's purest god the jewel-stones;
       But shapes and echoes that are never done
       Will haunt the workshop, as regret sometimes
       Will bring with human yearning to sad thrones
       The crash of battles that are never won.

From Robinson's COLLECTED POEMS:  THE CHILDREN OF THE NIGHT, CAPTAIN CRAIG (Macmillan, New York, 1915)

Struggling -- and then, after a while, Knowing

      Most of my experiences with solving mathematical problems have been challenging at first -- but often, after I explore and collect my thoughts, a pattern emerges.  The notion of "difficult at first" is vividly expressed in the following poem (found in the anthology Against Infinity (edited by Ernest Robson and Jet Wimp, now available at various used-book sites).

Geometry Test     by Larry Rubin

Thirty minutes, we had, to prove the theorem.
For twenty I sat staring at circles,
My inner angles frozen
When nothing came out equal.
The bisectors I drew were tilted wrong
While fear of the circular face of time
Stiffened my blood like clock-hands
Tracing arcs I never knew existed.
Suddenly that curve stretched perpendicular --
Longer that my longest transverse line --
Reaching beyond the limits of the page;
And the tallest segments of the intersected cone
Slit the seal of infinity.

My mind was washed like windshields after rain
And circles glided smoothly into place,
The arcs connecting in their shrunken frames,
I left that room, all theorems proved.