ALDO’S DIARY by Frank Morgan

“Mom, Penny beat me at chess.” Penny’s mother, Elizabeth Murrow, stopped loading the clothes washer and stopped thinking about triangles in the hyperbolic plane.

“Penny?”

“Yes, I did — I beat Aldo at chess. And he’s two years older than me.”

“But I thought you said you weren’t any good at games.”

“I didn’t think I was.”

“Aldo, did you let her win?”

“She forked my king and my rook.”

“Penny, you forked his king and his rook?”

“Yes, I planned it, like Dad did to me last night. I attacked his king and his rook at the same time with my knight, and since he had to move his king, I got his rook.”

]]>A dusty pedestrian road by the Alhambra is lined with engraved quotations. My favorite on Saturday was by Thoreau, translated into Spanish. Here are my own English version constructed from memory as I continued my walk, followed by the English original which I later found on the web, and finally my revision.

My version:

All of our excursions end by finally coming back to where we started. Half the journey is returning home. Sometime we will learn to extend our little walks by desiring never to return. When you have said goodbye to your parents, your wife, your children, and your friends, never to see them again; when you have paid your debts, written your will, put all of your affairs in order, and are a free man; only then are you ready to take a walk.

The original, “Walking” by Henry David Thoreau

Our expeditions are but tours, and come round again at evening to

the old hearth-side from which we set out. Half the walk is but

retracing our steps. We should go forth on the shortest walk,

perchance, in the spirit of undying adventure, never to return–

prepared to send back our embalmed hearts only as relics to our

desolate kingdoms. If you are ready to leave father and mother,

and brother and sister, and wife and child and friends, and never

see them again–if you have paid your debts, and made your will,

and settled all your affairs, and are a free man–then you are

ready for a walk.

For more of this essay see http://tanaya.net/Books/wlkng10/, e.g.

“It requires a direct dispensation from Heaven to become a walker.

You must be born into the family of the Walkers. Ambulator

nascitur, non fit.”

or “what would become of us, if we walked only in a garden or a mall?”

**Analysis.** His first sentence is better, more specific. His second is weakened by exaggeration, since we rarely actually “retrace our steps.” My third sentence is a mistranslation, but much better from the start, and infinitely better in omitting the ridiculous embalmed hearts and desolate kingdoms. In the fourth sentence, my “when” is much better than his “if.” Here’s my revision:

All of our excursions end by finally coming back at evening to the old hearth-side from which we set out. Half the journey is returning home. Sometime we will learn to extend our little walks by desiring never to return. When you have said goodbye to your parents, your wife, your children, and your friends, never to see them again; when you have paid your debts, written your will, put all of your affairs in order, and are a free man; only then are you ready to take a walk.

]]>Fortunately on the last Hand 24 I competed to 5Sx with everyone else in game the other way for 15 IMPS to tie for first.

]]>I teach mathematics as an adjunct instructor at Berkshire Community College (BCC) in Massachusetts, which is one of a handful of states that does not allow its employees to pay into Social Security. To get my last requisite credits toward Social Security, I took a part-time job at Home Depot. Home Depot has taught me a lot about gratitude and dignity.

I have met several people at Home Depot who are good, smart, hard working people. I go to them often when I have an issue I can’t solve on my own. Some are part time like me, working unpredictable hours at $11 an hour. I doubt even the few full-time people earn a whole lot more. (Just like BCC, Home Depot tries to minimize full timers in order to avoid paying benefits, though Home Depot does offer some benefits to part timers if they work there long enough.)

I work there by choice, for a special reason, but for these people. This is their life. One girl has no car, so she has to take the bus to work. But if she is scheduled on a day when no bus service is available, she has to take a cab to work. If she’s scheduled for four hours at $11 an hour and a taxi costs $20, then it’s almost like she’s working for taxi money.

I have learned a lot at Home Depot, but not what I expected to learn. All my A’s in college were worth nothing here. I had to learn from a new set of teachers. I think about the time Kathy had to teach me the proper way to water because I was drowning all the plants. I think about Dennis, who patiently taught me how to use a pallet jack. How he refrained from laughing at me during the fifteen minutes it took me to awkwardly move a pallet of merchandise ten feet is a mystery to me. I remember his yelling at me for “fluffing”— pretending to work by moving stuff around on the shelves. He and Ian (who I swear has experience with every product in the Garden Department ) then proceeded to show me how to use the phone and the giant ladder to stock properly. I think about Wayne, a man of few words, who I followed around like a puppy dog because he knew all the day-to-day things that I needed to learn how to do. These were my teachers . Yes, they taught me how to do my job. But the real value of what they taught me had nothing to do with Home Depot.

I see them do their work with a quiet modest sort of dignity that I think is sometimes lacking in academia. I learned to respect the grizzled old guys and the world-wise young ones for being experts at what they do and working a whole lot harder than I have ever had to work in academia. But I guess humility and strength of character is where you find it. I think of Ed Clark, who was head of the Engineering Department at BCC when I was a student. He always made it a point to get to know the custodians and secretaries and security. He knew their names and cared about their problems. That always mystified me. To me, they were like invisible machinery, necessary, background noise in the service of students and teachers and administrators. I failed to care about or even see their dignity and their humanity.

Sometimes I gripe in my head about how BCC has treated me. But my job is something the people at Home Depot can only dream about. I think about how overjoyed they would be to have what I have. I think about how much I take for granted.

I think about what I have read, that these are the type of people who support Donald Trump. But doubt he has even a clue about the financial problems these people face every day, their real, day-to-day struggles. I think he is just using them to further himself and his agenda.

I am luckier than I have ever known. God had been far better to me than I have ever realized. I can only open my eyes and say a little prayer of gratitude and wonder why I am so lucky. I believe there is a purpose in everything. I am coming to know the purpose of my summer working for Home Depot.

]]>**Irrational Exponents in Fermat’s Last Theorem. The n > 2 case.**

by Nicola Marino

In this paper we find explicit irrational exponents greater than 2 solving the Fermat equation, which we call Algebraically Derived Ratio of Irrational (or ADRI) numbers. We then further expand the class of ADRI numbers to cover all the solution exponents for any Fermat equation.

]]>**Theorem** (Euler). *A pseudograph has a circuit containing all edges and vertices if it is connected and every vertex has even degree.*

Proof by induction on the number *n* of vertices.

Base case *n*=1. Just follow the loops in succession.

Now assume for *n* and prove for a pseudo-graph of *n*+1 vertices. Pick a vertex. Since degree even, you can pair the incident edges, and you can avoid pairing the two ends of a loop. Short-cut each pair to avoid the vertex and delete it. By induction, each component of the new pseudo-graph has the desired circuit. Now restore the vertex and undo the short cuts to obtain the desired circuit.

by John Berry, Matthew Dannenberg, Jason Liang, and Yengyi Zeng

2015 NSF “SMALL” undergraduate research Geometry Group

With update below by Iglesias-Ham (all already known).

**Abstract**. The classic result about the optimal hexagonal packing of unit disks in the plane has recently been partially generalized by Edelsbrunner et al. to allow but penalize overlap for the case of lattice packings. We attempt to remove the restriction to lattice packings.

**1. Introduction.** Recently, Edelsbrunner, Iglesias-Ham and Kurlin [EIK] considered a relaxed packing problem in which disks are allowed to overlap and the goal is to maximize the probability that a random point is contained in exactly one disk. They found that among all relaxed lattice packings of congruent disks in **R**^{2}, a regular hexagonal packing with disks of a unique radius maximizes this probability. We attempt without success to remove the lattice hypothesis by generalizing Thomas Hales’s proof [H1, H2] of Thue’s Theorem, which is based on an idea of Rogers [R]. The key to Hales’s proof is to partition the plane into three regions and show that the density of disks inside each region is no more than the hexagonal packing density, but when we allow disks to overlap, the density of disks inside one of the regions is larger than the relaxed hexagonal packing. Acknowledgements 1.1. This paper is the work of the Williams College NSF “SMALL” 2015 Geometry Group. We thank our advisor Professor Morgan for his support. We would like to thank the National Science Foundation, Williams College, and the MAA for supporting the SMALL REU and our travel to MathFest 2015.

Continue reading paper**.**

Dear Frank Morgan,

Recently, while working in our camera ready, we learnt about some old results relevant to our paper:

– Fejes Toth in his book ‘Regular Figures’, 1965, posed the very same version of relaxed packing we address in our paper. He conjectured that indeed the regular hexagonal lattice would provide the optimal configuration when the radius is such that disks intersect each of the six neighbors in 30 degrees arcs. Moreover, he claims Bal’azs proved it for lattices in an unpublished paper back then.

– Bal’asz, 1973, published an article with the proof of the result for 2D lattices (“Ueber ein Kreisueberdeckungsproblem,”Acta Math. Acad. Sci. Hungar. 24, 377-382).

– Blind, G. and Blind, R, in 1986, proved the regular hexagonal lattice is still optimal among any distribution of disks. (“Ein Kreisueberdeckungsproblem,” Studia Sci. Math. Hungar. 21, no.1-2, 33-57 ).

– Blind, G. and Blind, R, in 1994, studied the same optimization problem for disks lying on a sphere (“Ein Kreisueberdeckungsproblem auf der Sphere,” Studia Sci. Math. Hungar. 29, no. 1-2, 107-164).

I thought as you have students working on the topic this information may interest you. Unfortunately, we have been working on a solved problem.

Best Regards,

Mabel

*Thanks to Iglesias-Ham for allowing us to reprint her message here.*

Since Bellevue College is open-enrollment, we see an incredible range of students. Some have been academic failures their entire lives and some are just using Bellevue College as a stepping-stone to elite universities and technical careers. I think this diversity makes it interesting.

**Some of my memorable students.**

1. 20-something Hispanic male. Parents disowned him when he came out as gay in his late teens. Spent a couple years couch-surfing and working menial jobs. Tried a couple community colleges but dropped out. Was at the top of the class in my Calculus III and IV classes, graduated from the University of Washington with honors in neurobiology. Joined Teach for America and is applying to medical school.

2. Late 20-something black man. Was working full-time as a fireman and EMT and taking classes aiming for medical school. Top of the class in Calculus IV.

3. 30-something woman. Was married and growing organic crops in California when her 10-year marriage ended unexpectedly. Came to Bellevue to stay with her father and took some classes for fun. Top student in Calculus II and IV. Graduated from the University of Washington in Computer Science and with high honors in mathematics (just because it was fun). Works at Microsoft.

4. Late 20s. Major HS activity was her boyfriend. They married and when their child started school, Cindy started taking classes at Bellevue. Good A- student in Calculus I and II. Earned a bachelors degree in Computer Science at the University of Washington.

5. Late 20s, 4’s friend and study partner. Worked the midnight to 8 AM shift stocking groceries. Graduated from the University of Washington and teaches AP calculus at a local high school. Students tell me she is tough.

6. Late 30-something nurse. Made money selling a medical patent and came to BCC to follow her passion to be an astronomer. Hard-working student in Calculus I and II (great role model for the younger students), graduated from the University of Washington in astronomy. Set up a scholarship at BCC.

7. Early 20s. Had a band in high school and wanted to be a rock star. Realized after several years of working in a print shop that the band was going nowhere. Came to BCC to take some math classes. After finishing all of the calculus courses with top grades he was seduced by physics. Was a top graduate in physics at the UW and got a PhD in physics from Princeton. Now a professor at a very-selective private college and is editor of a national physics journal. Contacted me several years ago beginning You probably don’t remember me .. But I did remember.

8. 20-something. Got a job at an aerospace company right out of high school as an engineering tech. Specialized in inertial guidance systems and trained the newly hired engineers. Realized the people he was training made 3 times his salary so he came to BCC. Earned an Electrical Engineering degree at UW, stayed at his company and made a lot more money.

9. 16 year old high school drop out – he didn’t fit in hs. First generation college student. Took the entire calculus sequence at BCC, worked overtime at the bicycle shop to save money for the Budapest Semester. Degree in mathematics from UW, another degree in Computer Science. Works at Microsoft.

**High School students** taking mathematics at Bellevue.

These would have succeeded without community colleges but we helped them along.

10. Stanford BS and MS in Mechanical Engineering. In a conversation with her grad school advisor, she is really smart, even by Stanford standards. Works in San Francisco for daVinci Robotics doing medical robotics.

11. BS in math at MIT (was also accepted at Harvard and Princeton), now in the doctoral program in math at Berkeley. Honorable mention in the Putnam.

12. Degree from Cal Tech. She is now in the doctoral program at Stanford in Ecological Systems.

13. Degree in math from Duke. Elected as the student representative on the Duke Board of Trustees. Masters in Statistics at Harvard.

14. BS and MS in Computer Science from Stanford. Now a VP of product development at a Silicon Valley start-up. Honorable mention in the Putnam.

15. Degrees in mathematics and economics from the University of Chicago.

16. BS and MS in engineering at MIT. Now in the doctoral program there.

17. BS in engineering at Harvey-Mudd. Now works locally in computer science.

18/19. Harvard and Wharton in finance. Now work for big east-coast firms.

10. 17 years old at BCC. BS in math and philosophy at the UW, PhD in math at Dartmouth. Now chair of a university mathematics department.

**International students:** Many great math students come to Bellvue to bring up their English proficiency before transferring to a university. One Multivariable Calculus class of 37 students had students from 19 different countries who had 17 different first languages (including a couple African languages I had never heard of). Bellvue has students from 60 foreign countries, mostly Asian, middle eastern and eastern European.

**Veterans:** I’ve had several veterans who began with beginning algebra and 9 quarters of mathematics later were in Calculus IV. Not top students at that level, but algebra to multivariable calculus in three years takes lots of hard work and grit.

**More**: I have also taught the lower level algebra classes with lots of first-generation college students, single mothers, people in drug and alcohol rehab programs, and people with lots of other personal issues. Many really nice people trying to get an education, but with very difficult lives.

**Faculty**: Even our math faculty have some interesting stories.

Berthe’s Jewish family had to leave Egypt in 1956. After a couple years in Europe the family came here, she studied at Bellvue, got a masters in math at the University of Washington and joined the BCC math faculty. Now retired, but endowed a math scholarship here for women.

After the revolution Tony left Iran by mule over the mountains to Turkey and then to a camp in Europe. Took classes at Bellvue, got a masters in math at Western Washington State University and joined the Bellvue math faculty.

Larry (the MC at the conference) was a high school drop out who then worked menial labor jobs for a couple years (even spent a week in jail in Salt Lake City as a 17-year old vagrant). Started at a community college, finally earned a PhD in math at the University of Minnesota and joined the Bellvue math faculty.

We also have tenured math faculty from Korea and India as well as part-time faculty from several other countries including a Jewish fellow from Russia with PhDs in math and physics.

And lots of bright young full-time faculty, our future.

Spring of 2002 was the beginning of the season that would make or break me as a painting contractor. I’d decided two years earlier to put my experience to work for me, and in some ways it was paying off. I was finally starting to charge what I was worth. But both the kids were still small, and I was constantly torn between their needs and my company’s. The mountain of taxes, bids, payroll, invoices and all the other things that go with small business ownership was becoming an avalanche.Painting was what I knew, and it felt like the only way I was ever going to make a decent life for myself and my children. Even now, I am a practical person: I accept what’s proven to me. At that time if someone had told me I needed an angel I’d have laughed. Up until then I had no evidence that angels even existed, but it took only one encounter to change my mind.

It was sunny, clear and warm, the kind of day at the beginning of the season that smells like money to painters. The first few hours of the morning had been spent on phone calls, gathering tools, getting youngsters off to daycare and workers off to jobs. I left the house late and realized I had absolutely nothing to eat. So I stopped at QFC, and when I got out of the store with an armful of sandwiches and a mouthful of barbeque burrito, someone was waiting by the car.

“I used to be a painter too.” She told me in a rather sweet, high voice. It seemed an odd way to start a conversation. Maybe she’d noticed my smeared clothes or the car full of tools. I don’t recall what she was wearing, but she had dark hair like mine and was at least as tall as I am. She kept talking to me as I opened the car and put my lunch inside. I’m sure I replied to her; I’m a gregarious person and never let a chance to talk get by me, but I only remember the things she said to me.

“When I was thirty five I went back to school. That was four years ago. I just graduated with a Bachelor’s in Nutrition from Bastyr.”

I was thirty five.

She talked more about how glad she was to have a degree, how wonderful it was to learn about nutrition, how she wasn’t going to have to paint any more. She gave me her name and her phone number.

QFC is in the same parking lot with the Burger King that used to be Herfy’s. Hearing her story there was powerful. Back in ’82 Herfy’s had been my first job, back when I was an honors student at Nathan Hale High School. None of the shiny dreams everyone else had for me had survived. Anything less than an A was a failure in my dad’s eyes, and he pulled me out of high school in my senior year. I finally graduated two years late and just drifted away from community college and landed in the trades.To this day that parking lot reminds me of high school and being fifteen years old.

Her voice spoke to that girl who believed everything was possible. Somehow she made me see that time is all one piece. The parking lot was the same, the world was the same, I was the same. Nothing was left behind. All those possibilities remained. We talked about painting and Bastyr, not Nathan Hale and Herfy’s Burgers, but everything she said woke me up more. Unfortunately we couldn’t stand there forever, even though the morning was fine and the conversation was compelling. She went her way and I drove off eating the rest of my burrito.

The next week I signed up for an algebra class at Edmonds Community college. Within a year I was taking twelve credits per quarter.

The path that she set me on wasn’t an easy one, but it was the right one. Raising children as a full time student isn’t any simpler than raising children as a business owner. In the years that followed, all the stubbornness that allowed me to succeed as a female painter channeled into overcoming a different set of obstacles. Apparently angels don’t solve problems for us; they show us the ones with the most fulfilling solutions. At least that’s what mine did.

We spoke only one other time. She knows I went back to school and that I’m glad I did it. I lost her number, so I have no way of knowing if she remembers me, or if she realizes she’s an angel and that the story she shared with me literally transformed my life, but I’d bet she does. She knew what she was doing when she waited there by my car. That’s an angel’s job.

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