As a young boy growing up in West Virginia, in the early 1970s, my family would take these road trips across the state. We went everywhere, usually with a Scotty camper in tow. Dolly Sods, Blackwater Falls, Bluestone River โ everywhere. One day we were on another adventure when we passed a sign telling us to โTURN OFF ALL RADIOSโ and as we topped the hill, we saw a valley filled with HUGE radio dishes. As an autistic child, I lost my damn mind, โCAN WE GO? CAN WE GO? WE GOTTA GO? CAN WE? CAN WE? PLEASE!!!!!โ
Frank Drake in front of the Green Bank Observatory’s 8G5-1 (Tatel) Telescope. the first telescope built on site and used in the first SETI research โ Project Ozma
My grandparents, to their credit, wilted to my demands and we took a tour of the Green Bank Radio Observatory. On the bus, we heard the tour guide, โthis dish has a surface area of more than an acre,โ โthis is the dorm where the visiting scientists stay,โ โthis is the ball field where we wind down on weekends.โ The ball field was filled with deer that day. The last thing she mentioned was that a visiting scientist was waiting for us in the auditorium and he had an educational film to show the group. The scientist, speaking with a distinct accent, gave a short lecture – setting up the premise for a film on SETI โ the Search For Extraterrestrial Intelligence. He explained how our powerful radio waves have escaped Earth and continue out into the universe. Then he flipped it, โwhat if we listen for the signals that alien species might be broadcasting?โ
In those days I was already neck deep in Ham Radio and I damn near lost my fucking mind when I heard his thesis. After the lecture, after the film โ when the tour was over, I was a bundle of questions. The scientist patiently answered all of my questions. The last question (I was a dumb young kid from West Virginia), was why he stopped working in the hospital and decided to search for aliens? He smiled and explained โI am not a medical doctor, you can be a doctor in any subject.โ I enthusiastically replied, โIโm promise I am going to get one of those!โ I had no idea what I was talking about.
Years passed. My family soon moved to North Carolina. I was no longer serving time in the West Virginia Public School System. North Carolina education was light years ahead of West Virginia.
During high school, I had heard of this show on PBS titled โCosmosโ. I knew it was about space and that was about it. I geared up to watch the first episode. I had my cassette recorder, microphone, notepad and several pencils all set to record the knowledge of the universe there in our living room. And here came that man with that distinctive accent talking about the wonders of the universe! It was the guy! It was the guy that I met years previous at Green Bank. Wow, he was on TV! “I guess he finally made something of himself” said the dumb young kid from West Virginia.
I finished high school and charged my way through college. I took calculus-based physics back in 1984 and have been pissed off ever since. There were two brutal lessons I disliked in severe kinds of ways.
The first was entropy and the heat death of the universe. It wonโt affect me directly or indirectly, Iโll never see it – it is something I could have understood and went about my life. But nope โ a major depletion of serotonin and dopamine creeped into my autism-riddled brain. Greatly depressing.
The second one REALLY pissed me off and it is what pushed me down this path. In a greatly oversimplified description, in physics, we can know the velocity and vector of a particle, but we canโt know the temperature. Or, we can know the temperature and velocity, but not the vector. I get the realities of observational thermodynamics, but that doesnโt mean I am happy about it.
There has to be a way to know all three โ there has to be a path to figure all of this out. We are taught that we are NOT able to know all three, but nature knows. The fabric of the universe canโt exist if nature doesnโt know. Physics would cease to work if nature was clueless about the properties of all objects. NOT knowing is an issue.
NOT.
Hmmm, โNOT.โ That reminds me of something.
There is an idea there maybe -> NOT, NOR, AND, NAND plus many others added to calculus might create some interesting results.
At the time I took the physics classes, I had been studying Boolean Logic and circuit design for three years. We canโt come up with a way to know all three, but maybe, just maybe, there may be a way to narrow the answer down to where these values are NOT. And thus my exploration of locative calculus (Loculus) began.
Since those sad and angry days of 1984, I have continued working on Loculus โ on and off, mostly off, ever since.
The first step was to define Boolean Look-Up Tables (BLUTs) and they were a nightmare. All my ideas seemed like they were fighting me. If I change this, the model changes to that and melts down in a white-hot mess resembling Satanโs taint.
โIโll just set this aside for a bit,โ I would say after the panic attacks subsided.
This level of pearl-clutching, neurodivergent behavior would continue for decades. In 2022, I started on my fifth college degree, BS in Electrical Engineering. I just needed a few classes to complete that chore โ I hated it. Online education is the worst. Donโt even get me started on online testing proctors that will only speak to you in Arabic.
Yet, I survived and graduated while swearing that I am done with academics. I warned my friends and coworkers, โif you ever see me on another college website, I have one word for you โ chloroform.โ
Bastards let me down.
I keep pen and paper by my bedside in case I dream something noteworthy โ long story. Suffice it to say, a couple of months after I finished my last EE class, I must have had some sort of crazed dream because I woke up with this written on my nightstandโฆ
โฆ and on that gayly pink post-in note was the key I kept missing, the Loculus Object. After furiously writing day and night for who knows how long, I now have seven autism-powered papers pre-pub ready, I have begun looking for PhD By Publication opportunities.
IF I get these papers successfully published, and IF I find a school willing to listen to my madness, and IF they bestow a degree upon me, I will make a road trip to Ithaca, NY where Carl Saganโs headstone lies. There I will recite my Loculus abstract.โจโจ
That will close the chapter on the promise I made to Carl, all those many years ago.
That turns out to be the key to the whole of my idea of locative calculus; Loculus for short.
I finally have the outline of the 5 part series of research papers.
Loculus I โ Foundations and Arithmetic of State-Bearing Variables Loculus II โ Algebra, Functions, and Domains in State-Bearing Mathematics Loculus III โ Geometry, Trigonometry, and Vector Analysis in State-Bearing Mathematics Loculus IV โ Differential and Integral Calculus of State-Bearing Variables Loculus V โ Ordinary Differential Equations and Dynamical Systems in State-Bearing Mathematics
Anyway, writing continues on my PhD – I need something to do after work and on weekends.
When I finished my E.E. degree earlier this year, I told all my friends, โif you catch me looking at another university website, I have one word for you; chloroform.โ I was so beat down by my last degree experience, I never wanted to register for another class, like ever.
Well, that shit didnโt last.
Previous to the E.E. degree, I was working on a math PhD via PhD By Publication. All the cool kids are doing it. But the layoff I barely dodged in 2023 told me I need to add an engineering degree to my resume if I want to stay in commercial space. So, I backed off the PhD, and buckled down on the E.E. degree.
I have been recovering from that unreal push to finish engineering for some months now. I got clear of all the crap I had going on and now my future roadmap is empty as far as the eye can see.
Then I went to bed last Sundayโฆ
And woke up staring at a Post-It note I had written during the night. It contained the answer to the roadblock for my thesis that I have been fighting since 2022.
I kept it in my pocket all the next day.
It sat in thereโฆ
โฆ whispering to meโฆ
โฆthe secrets of the universe.
By the end of the day, I wrote 21 pages of notes. I got further that day on my PhD then all the previous time I had accumulatedโฆ from a Post-It note written during what must have been some sort of Saganesque fever dream of Carl battling flat-earthers from beyond the grave.
Things have moved quickly since. I have reconstructed the basic mathematical concepts, rewritten the first 5 papers, and already have the the outlines for two adjacent papers done.
Just a brief note on my academic progress. I postponed my PhD By Publication work to finish up my 5th degree โ B.S. Electrical Engineering. That on top of my day job in commercial space keeps me too busy. This has to be one of the least enjoyable times in my academic career. I have difficulty seeing the day-to-day grind as anything but a long slog.
I look forward to next year when I am clear of all the things I have going on.
Buying a house
Building a house
Moving
I have storage containers in three states that need to be shipped to the new house
Unpack everything
Finish my EE degree
Finish my Avionics MRB training at work
Plus add in all my photography nonsense
Gotta get clear of it all. It is obvious that I am biting off way more than I can chew this year.
Waaaaayyyy back in the day I used to be in advertising, like 20 yearโs worth of it. Mojo Zoo was the first ad agency I ever started. It was like school with many lessons resolutely learned.
More recently, I have been thinking about what to do post-retirement from the commercial space industry. Photography has always been my first love. I have always been passionate towards the arts, photography doubly so. If you look around my house today, it will look like a 1980โs vintage camera store. Even myy refrigerator has a drawer filled with film.
Oh yeah, as much as I have embraced digital, my roots are certainly in analog.
Post retirement, I want to have a photography studio to pursue my experimental style of fine art photography. But since I am so passionate about it all, I am getting a jump on my post retirement activities.
Welcome to the 232nd Carnival and my fourth time hosting the event.
To see past entries in the Carnival Of Mathematics and future scheduled hosts, please visit The Aperiodical.
I am honored to again host the Carnival of Mathematics! I learn so much from hosting; things I usually wouldnโt be exposed to are jam packed into every Carnival Of Mathematics post. Be sure to dig into the archive!
Here are the entries. Enjoy!
From Alice To The Mad Hatter: The Road To A PhD
By Storm Bear Williams
Everyone’s path to a PhD is different. Mine is more madcap than others.
This post is about the bottom center panel, โGame Theory final examโ.
I don’t know much about game theory and I haven’t seen any other discussion of this question. But I have a strategy I think is plausible and I’m somewhat pleased with.
(I assume that answers to the exam question must be real numbers โ not โโ โ and that โaverageโ here is short for ‘arithmetic mean’.)
First, I believe the other players and I must find a way to agree on what the average will be, or else we are all doomed. We can’t communicate, so we should choose a Schelling point and hope that everyone else chooses the same one. Fortunately, there is only one distinguished choice: zero. So I will try to make the average zero and I will hope that others are trying to do the same.
Iโve just uploaded to the arXiv my paper โPlanar point sets with forbidden 4- point patterns and few distinct distanceโ. This (very) short paper was a byproduct of my recent explorations of the Erdรถs problem website in recent months, with a vague emerging plan to locate a suitable problem that might be suitable for some combination of a crowdsourced โPolymathโ style project and/or a test case for emerging AI tools. The question below was one potential candidate; however, upon reviewing the literature on the problem, I noticed that the existing techniques only needed one additional tweak to fully resolve the problem. So I ended up writing this note instead to close off the problem.
Editor’s Note: Terance’s website if loaded with his work in mathematics.
Mathematics With Friends
by Ioanna Georgiou and illustrated by Asuka Young
“Mathematics with Friends” is a blog that takes a maths-related dialogue from the Friends TV series every month and explores the maths mentioned further. It comes with its own characters and amazing illustrations.
Books That Are About Maths, But Not About Teaching It
By Tom Briggs
In the spirit of getting straight to the point without providing my entire life history, please click below for a list of books that are about maths, but not about teaching it. Read beyond the link if you’d like a bit of context.
The list has been put together at The StoryGraph[2]: it can be browsed without creating an account and it can be searched, sorted and filtered, which I thought would be helpful as there are over 160 books.
A respectful story about the passing of a colleague. These things are never easy. Also an interesting correlation with Paul Halmos, the Hungarian-American mathematician. The full story is in the link below.
Math teacher Ben Orlin writes and draws the (aptly named) blog Math With Drawings and is the author of a new book, Math for English Majors: A Human Take on the Universal Language. To mark its publication, he devised this entertaining accompanying quiz. You can read the Ars interview with Orlin here.
Math, we like to say, is math. It often seems that mathematics is something beyond our controlโsomething unyielding, universal, inhuman.
But is it really?
Yes and no. Math may be the language of the cosmos, but itโs also a human language, with its own quirky etymologies, classic literature, and hilarious regional accents. In fact, you could say that math has as many stylesโas many idiolectsโas it has speakers.
Viterboโs conjecture was refuted by Pazit Haim-Kislev and Yaron Ostrover
by Gil Kalai
Claude Viterboโs 2000 volume-capacity conjecture asserts that the Euclidean (even dimensional) ball maximizes (every) symplectic capacity among convex bodies of the same volume. In the recent paper A Counterexample to Viterboโs Conjecture, Pazit Haim-Kislev and Yaron Ostrover disproved the conjecture.
In a dimly-lit tavern on the South Bank of the Thames, Philip Henslowe โ builder and owner of the Rose Theatre โ is celebrating the success of Shakespeareโs latest blockbuster, Henry VI Part I, among the cutthroats, actors and other lowlife of London. He spreads thirteen playing cards on a table in a circle. โPick a card,โ he grins. โAny card.โ
And this article is complete with cryptic, vintage explanations!
Editor’s Note: Now I want to visit the Dulwich College Archive!
Book Review: The Best Writing on Mathematics 2021
by Ioanna Georgiou
Mircea Pitici (Editor) PRINCETON UNIVERSITY PRESS 2022, 320 PAGES PRICE (PAPERBACK) ยฃ20.00 ISBN 978-0-691-22570-8
This book contains articles written in 2020. The editor speaks of the difficulties accessing libraries and resources, something we are all a bit too familiar with. Here I will comment on each of the twenty-six articles selected as the โbestโ and draw an overall conclusion at the end.
Editor’s Note: We have to start doing a better job in archiving and making available the collective of human knowledge to everyone. The defunding of libraries has to stop. The abusive research publication industry has to stop.
In college, homework assignments generally make up 5-20% of your grade, but can be the biggest time-suck for most students. Yes, working problems is one of the best ways to turn new concepts into working knowledge, but a large majority of those problems that take you hours and hours to work through, youโll never see on an exam.
Itโs like an automatic reaction. New concept = go to Google for a quick explanation. Donโt think just because your professor gives you a textbook and some examples on the blackboard that youโre limited to that information. You have a massive free search engine at your fingertips, so make use of it.
5. They study in short bursts, not long marathons.
Studying in short bursts tends to help you focus intensely because you know there is at least a short break coming.
This also fits in nicely with our Ultradian Rhythm, the natural activity/rest cycle of our bodies, which makes studying continuously for multiple hours on end counterproductive.
6. They reverse-engineer solved problems.
Itโs one thing to follow and memorize a set of steps to solve a calculus problem. Itโs an entirely different thing to understand what a derivative is, be able to take derivates of complex functions, know when to use the chain rule vs. the product rule, etc. The problem with simply following the steps the professor provided, or the textbook outlines, is that youโre only achieving a surface-level knowledge of the problem. Top students, instead,take solved problems and work backwards, from solution to question, asking โwhy.โ
Why did this get this value? Why did they simplify this expression? Why did they use that type of derivative rule?
By following this process, you begin to understand the interconnections of the concept, and how to directly apply that to a problem. This โworking knowledgeโ of a concept is key to performing well on exams, especially on problems that you havenโt seen before.
7. They donโt own a highlighter.
Highlighting anything = unengaged reading. If you want to note something that stands out, underline and write a corresponding note to go along with it. Or better yet, write yourself a note summarizing the item in your own words.
8. They sleepโa lot.
The daily routines of top performers, in any field, are characterized by periods of intense work (4-6 hours per day) followed by significant quantities of high-quality sleep (9 hours per night). You see this trend in top violin prodigies and chess champions, as well as elite athletes. The idea is to alternate periods of intense work with rest, so that you create tons of new connections in your nervous system, and then allow adequate time to assimilate those gains.
9. They engage themselves by asking questions.
What happens if I tell you, โThomas Jefferson almost single-handedly drafted the Delcaration of Independence in 1776.โ?
You might say โHmm.. thatโs interestingโ, try to remember it for later, maybe even write down a note or two.
But what if I ask you, โWho was Thomas Jefferson?โ What changes?
You start searching your memory, sifting through images of old guys, founding fathers, thinking about the Declaration of Independence. You come up with your own narrative, and then realize that you have gaps.
When was he around again? And why was he so important?
Youโll probably find yourself going to Google to fill in the gaps. Through that process your learning will be much more deeply seated in your brain than anything your history teacher ever told you about him. Thatโs the power of asking questions.
10. They make the best out of lecture.
Yes, your professor sucks. Yes, lectures are boring. Yes, itโs either too fast so you canโt keep up and miss all the important stuff, or itโs way too slow and you start zoning out because you already understand everything.
The best students look at this this way: Iโm going to be there no matter what, so whatโs the best use of my time while Iโm in the classroom? Ask questions, bring the textbook and look stuff up, focus on the important practice problems to copy down in your notes, try to anticipate what the professor is going to say, make note of anything they put emphasis on as a potential exam topic. All of these things make the time you have to spend in lecture more productive and engaging. And thatโs less time you have to spend studying later on.
11. They over-learn.
School is hard enough, with the amount of studying and homework you have to do. And on top of all of that Facebooking you have to get done? It might seem ridiculous to suggest learning more than you have to.
What!? Are you insane!?
But this is precisely what top students do. And paradoxically, they end up spending less time trying to understand how to do homework problems, andless time studying for exams because of it. Because when you โover-learnโ past whatโs presented in class, you build a better framework for the subject.
Think of trying to remember some details about Abraham Lincolnโs life. You try to remember the dates of the Civil War, or what he said in the Emancipation Proclamation. You study the same facts over and over and over againโฆ but itโs just boring, and you quickly forget. But what if you knew his whole lifeโs story? About how Lincoln suffered from bouts of depression, and his relationship with his wife suffered? You start to learn that the dude was human, and you start to relate to the things he did and the struggles he went through. Now youโve constructed a story in your head. And studies show that humans learn best through stories. So yes, itโs more information, but your brain knows what to do with it now that all those random facts are linked together. More learning, but less rote memorization and struggling to remember random facts.
12. They immediately study their exam mistakes.
Most students get their exam grade back, flip through to see if the professor made any mistakes they can argue about, and then promptly shove it into their notebook, never to be seen again until the mad scramble at the end of the semester to study for the final.
Instead, top students ignore what they got right, and use their mistakes as an indicator of what to improve on.
13. Theyโre busy with work and side projects.
Yes, to do well in a course, you need to focus and put in the hours. But like many geniuses throughout history have shown, involvement in a diverse set of subjects, activities, and skill sets keeps you active, and provides you with a rich and diverse set of mental models to pull from.
Also, as they say, โIf you need to get something done, give it to the busy person.โ If you stay active in multiple areas, you donโt have time to procrastinate, and are forced to be efficient with your study time. This generally translates into quicker learning and better performance throughout the semester.
14. They use lecture as a detective mission.
Though completely unaware of this fact, your professor has tells. Yes, like in poker. Tells during lecture will hint at particular types of concepts and problems that will be emphasized on the midterm or final exam. The best students pay attention to topics professors spend a seemingly inordinate amount of time on and make note. Chances are youโll see something related on the final.
15. They donโt wait for motivation to strike.
Motivation comes and goes, but studying for a degree requires persistence and consistency. Just like Olympic athletes train even on their worst days, the best students figure out how to get their coursework done when itโs the last thing they want to do.
16. They practice under test conditions.The old adage โpractice makes perfectโ isnโt totally true. Deliberate practice under the right conditions, with the correct mindset, is more like it. Instead of reading through all of the lecture notes and redoing old homework problems, top students make themselves practice exams, and rehearse their exam performance, under time pressure and in similar conditions (no notes, uncomfortable chair, quiet room, etc.) to what theyโll see on test day.
17. They use old exams.
Professors arenโt the most inventive folk. Along with coming up with lecture material and departmental responsibilities, theyโre also primarily concerned with research. So typically midterms and final exams more or less look alike for similar courses year-to-year and even across universities. Because of this, old exams are a gold mine of opportunity for figuring out what problems you should be able to solve and study from.
18. They make their own study guides.
The best students donโt simply use the study guide the teacher provides, they create their own.
Creating the study guide is half the battle, requiring you to go through your notes, consolidate them, and organize them in a way that you understandโall valuable study activities. Youโll also be able to use your equations sheet much more effectively on the exam itself (if allowed) because you know exactly where everything is.
19. They actually write on paper.
Writing out notes on a laptop is efficient. Too efficient. Because itโs so easy to quickly type out exactly what the professor is saying, you donโt have to do the work of trying to figure out how to consolidate the information into your own shorthand. Some also believe that the act of writing helps retain more information.
20. They use the 80/20 rule.
Yes, some students who get good grades do every reading assignment, finish every practice problem, and attend every study session they can get their hands on. But these students are missing the point. There will always be an endless amount of information you could learn given the time and effort, but having the ability to discern what is worth learning will truly set you apart.
Top students identify the 20% of concepts they need to learn deeply, in order to determine 80% of their final grade. They focus intently on those few things, and simply ignore the rest. This is a formula for high performance, without hours and hours of busywork. And it translates seamlessly into the real world too.
21. They donโt complain.
Complaining simply has no place in the smart studentโs repertoire. If something sucks, change it or ignore it, but donโt waste your time, energy, and mental state talking about it. Got a crappy professor? Either switch class sections or focus on teaching yourself. Horrible textbook? Find alternate resources (Google is free in case you hadnโt heard).
22. They learn by doing.
Any technical subject can only truly be internalized through use. Just like learning a new language, learning to be fluent in algebra or calculus requires active application of rules and formulas. Top students know there is a big difference between knowledge, and applied knowledge.
23. They take personal responsibility for learning the material.
The best students understand that they, and only they are truly responsible for their own education. So waiting to be spoon-fed by your professor and doing the homework assignments will never be enough. Despite your schoolโs best intentions, theyโll never be as committed to your academic success as you can be.
24.Following what they love
Those students you admire are passionate about what they are learning. They have the drive to develop their learning further based on their love of what they are discovering. This may not always be the case and is often unavoidable but if you follow what interests you and cultivate a curiosity of this area, your motivation to learn will thrive.
Not every student is the same and many top students donโt follow the status quo. The best way to create good habits for students is to try a variety of techniques and figure out what works for you.
25. Question your teachers Thinking outside the box is a cliche but certainly a reality for students. They question everythingโespecially test questions they get wrong. This attitude is important because it shows a general inquisitiveness that is essential in learning. As any parent of small children knows, questions are a way to gain knowledge. Teachers canโt be offended when a student asks a difficult question and parents should encourage this behavior.
26. They know the best way to study.
It is important to know the best way to study for yourself. Do you need pictures? Sounds? Study better in quiet or noisy environments? Figure out what works best for you so that you can make the most out of your study time.
27. They play hard.
We all know that a balanced lifestyle is the best way to stay mentally and physically healthy! Top students donโt spend all day in the library grafting (contrary to what you might think!) Rather, they take the time to put their studies aside and do something which is fun and exciting!
To see past entries in the Carnival Of Mathematics and future scheduled hosts, please visit The Aperiodical.
I am honored to again host the Carnival of Mathematics! I learn so much from hosting; things I usually wouldnโt be exposed to are jam packed into every Carnival Of Mathematics post. Be sure to dig into the archive!.
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