P-adics

When I was young, learning math(s), I would often wonder what more "advanced" mathematics (like my older brother and sister were taking) could be about.  At a young age, I remember actually coming to the conclusion that you learn to add, subtract, and multiply, with really big numbers!  I had no clue there could be so much more to it.

Maybe because of those early imaginings, some years ago I started playing around with a mathematical idea that's not dissimilar to doing basic operations with big numbers.  (Maths are great for thinking about when you're drifting off to sleep, since they can consume your whole brain and there's no room for the troubles of the day.)  I got pretty far with it, I think, and told a few people that I thought I discovered something, before coming to the conclusion that nah, it was all bunkum.  Here's a social media post I had started composing to talk about it (never posted), right after I gave up on it:

Disclaimer: I’m not a mathematician, but I’ve been playing with some back-of-the-envelope sort of stuff (actually done mostly while lying in bed at night, drifting off to sleep) and have made a tiny bit of progress on some (I’m sure) unprofound mathematics that probably wouldn’t have been a challenge for most of you.  But it’s been a lot of fun!  So I thought I’d write it up here.  And it all started with a misconception…

I began considering digit sequences where you know the low-order digits but not the high order digits -- for example, “...20445089241” is just some number that ends with those digits, but the “...” means that we don’t know how the number begins.  The misconception was, I began to consider these numbers as infinite -- that is, the “...” goes on forever.  So these numbers are all effectively infinity, but they just… end differently.  Knowing the final digits was enough to give them different properties -- for instance, divisibility.  For example, a number could be infinite but still may or may not be divisible by 5, depending on whether the last digit was a 0 or a 5.

(I’m sure you can all see the fallacy here, that any such infinite digit sequence is equal to any other, and there’s no divisibility difference between them, but pondering the possibility was enough to keep me going.)

So, treating them as a class of numbers, I began to ask the obvious next questions about operations and closures.  Adding two such numbers is trivial -- e.g. “...1230734 + ...6534221 = ...7764955”.  Subtracting two such numbers was doable, but not knowing which was “larger” meant it yielded two answers (A-B and -(B-A)).  Multiplying was trickier, but I managed to figure out the algorithm for generating the result starting with the least significant digit.  Division was a bit trickier still but I was still able to figure it out eventually (requiring an assumption that it divides evenly -- you don’t really know unless you know the higher order digits).  From the division algorithm I devised a way to figure out the square root of one of these numbers (again starting from least significant digits).

Then I had the odd thought, since these numbers were infinite, but I could still apply math operations to them, then could I find a number A that is its own square root (i.e. A*A=A, excluding 0 and 1)?  Turns out I could!  For instance, “...141376 * ...141376 = ...141376”.  This got me excited.  But a later Google search revealed that I had just rediscovered automorphic numbers!

It wasn’t until I started trying to devise a way to convert these numbers into different bases that it fully dawned on me that my premise was flawed -- all these infinite numbers were equivalent and didn’t have any unique properties; they’re just different representations of the same thing.  But all the methods for basic operations still work for finite numbers where you just don’t know the high-order digits of the number, so that’s pretty cool, to me at least.  Thanks for reading!

It was years ago that I basically stopped thinking about the idea.  But then recently, I saw a passing mention that p-adic numbers were numbers that go off infinitely to the left, and my ears pricked up.  So I went off and watched this video and it immediately clicked.  It really feels like I was close to (RE)discovering them!  I was wrong in dismissing them as "just" infinite numbers, and I shouldn't have just given up.  There were still a few places where I would have had to make a logical leap that I don't think I could have.

It turns out p-adic numbers were discovered in the late 18th century, and have been expanded by some top minds ever since, so I won't ever feel bad that I didn't flush out the whole system.  On the contrary, I feel quite exhilarated at the discovery and will do a bit of further play with them.  And I'll be putting p-adic numbers into my personal keep chest of things I (re)discovered, along with chart parsing and cyclical cellular automata.  Even if I blundered on to familiar territory, it still makes part of wish I had pursued mathematics more academically -- I would have had no chance at greatness I think, but there's millions of mathematicians who pursue it out of love and curiosity, and just to be part of the search for fundamental truths, which I think is noble.

This horrible year

2024 is drawing to a close, and good riddance.  We lost my dad in February; my mom, a month later in March; and Marjorie's mom just three days after that.  Needless to say we were in shock for a long while afterwards, but I wanted to include some things here for posterity.

Dad's obituary, from this site...

John Philip Schnitzius passed away on February 10th, 2024 in Cocoa Beach, FL, just shy of his 87th birthday.

John was born on 02/14/1937, the son of John Alphonse Schnitzius and Frances Rose (Szymanski), and grew up in Mount Ephraim, New Jersey. He acquired his associate degree from Temple University and his BS in Electrical Engineering from Drexel University. He met his life partner Loretta at the Dancette Ballroom in Oakland, NJ, and they were married on 02/24/1962. John worked as an electrical engineer for many years, for NCR, Harris Corporation, and NASA among others, living in Pennsylvania, New Jersey, Delaware, and Florida before retiring in Cocoa Beach.

John was an active boater and sailor for many years, and was a member of the Grumpy Ole Sailors and the East Coast Sailing Association. Whenever he wasn't on a camping trip with Loretta, he was planning his next trip. He loved adventure, fishing, hiking, camping and traveling. He was an amazing husband, father, grandfather, and great grandfather, and always had many friends, old and new. John has been described as wise and kind and humble, always willing to give advice or help with a lending hand.

John was a eucharistic minister and longtime member of the Church of Our Savior in Cocoa Beach.

He is survived by his wife, Loretta of nearly 62 years, his three children, Kevin Schnitzius and his wife, Kim, Danielle Schnitzius Norwood and her husband, Charlie, and Mark Schnitzius and his wife, Marjorie; his five grandchildren, Abi Schnitzius, Livia Schnitzius, Bryce Schnitzius, Chaeli Norwood Harden (Kevin), and Chase Norwood, and by his great grandson, Kix Harden. He is preceded in death by his parents, John and Frances, and by his sister, Clare, and brother-in-law, Bill Leap.

Services will be held Friday, February 23rd at 10:30 a.m. at the Church of Our Saviour in Cocoa Beach. Burial will follow at 12:30 p.m. at Florida Memorial Gardens in Rockledge. In lieu of flowers his family asks that donations be made in John’s name to the Cancer Center of Merritt Island.

The text of my Facebook post about Dad...

Dad was really into boats and fishing, which never really rubbed off on me. Even still, I have lots of great memories of family trips in the boat out on along the New Jersey coast. We also took a long driving trip down to Florida every year for quite a few years in a row. It was a happy childhood for me. He loved an adventure, which certainly *did* rub off on me, but only in retrospect do I think his main motivator was making his kids happy.

Dad always, always had time for my questions, and I had a lot of questions. Sometimes when I only wanted a quick answer it became an hour of writing things out on graph paper or looking things up in the Encyclopedia Britannica. He taught me about math and chess and computers, and from him I credit that part of me that takes problems aside that initially seem to be a complete mystery and sit with them until I figure them out.

Going into computers was an effortless career choice for me, and so it was no real coincidence we both ended up working at the same huge area employer of technical people of all sorts: Kennedy Space Center.  We worked for different companies there, but we ended up working in the same building (among many) for a year, maybe two?  On the father-son relationship timeline, this had to be a peak.  

He loved his music, but had no real ability, and I think deep down he would have loved it if one of us became a concert pianist. He loved a good joke, too, but had no real ability to tell one. He would sometimes try, then get to the punchline and say something like “Wait, I want to make sure I get this right. [pause] So, the *banker* says...”.  And where a lot of people say they use profanity to blow off steam, my dad did it better than anybody -- he hardly ever swore, but sometimes he would mess something up, flash with anger shortly, saying "Ohhhhhhhhhhhhh..." and end it with a quiet "shit" before breaking into laughter.

When one of his early jobs wanted him to start clocking in and out, he refused, and quit. But later on he took on a few jobs that had him commuting some really long distances so us kids could stay in our same schools with our same friends. I remember him as anxious frequently during these times, but as the years went on he came to realise his kids were turning out happy and successful, and he seemed to relax a lot. He of course wasn’t happy when I moved away, then further away, then even further away, but us kids were always free to set our own course.

Human lives can’t be summed up in a few paragraphs, so any obituary or eulogy is going to come up short. But I want to share a few little things that have come up in my memory this week that might give some glimpse into how *nice* of a man he was.

One time my parents came to visit while we were living in Singapore, and while they were there we took a slightly nutty side trip into Malaysia. On the long bus ride back I sat a few rows behind my father, and sat fascinated as he chatted with the tour guide across the aisle. The guide was Malaysian, Muslim, and probably 30 years my dad’s junior, and having nothing in common and without artifice, my dad just kind of effortlessly befriended him, across the cultural gap, and they shook hands warmly and said goodbye at the end of the trip.

Another time I came home for a visit, my dad mentioned to me how he had sold the old electric guitar we had in the attic at a garage sale, for like $5. It came to light that he had told the person who bought it that it worked; I told him though that it was actually broken. I started to make a joke about it, but then I saw how crestfallen he was – it wasn’t enough that he thought he was telling the truth at the time, it had to actually *be* the truth. And so we sat there a while brainstorming ways to get back in contact with the person.

I could go on. But it’s inevitable that most of my personal memories of my dad will die with me, as the memories of his dad died with him. Every morning since his passing I’ve woken up to thought, wow, it’s really real, Dad is no longer in the world. I have a lot of friends who have known grief in their lives, some all too recently, who must know how this is. I just really, *really* would like to talk to him again, and not even about anything.

Mom's obituary, from here...

Loretta Eleanor Schnitzius passed away on March 12, 2024, in Cocoa Beach, FL, just a month after her husband John of 62 years.

Loretta was born on October 28, 1937, the daughter of Felix Michael Thompson and Helen (Gąsior), and grew up in Camden, New Jersey. She met her life partner John at the Dancette Ballroom in Oaklyn, NJ, and they were married February 24, 1962.

Loretta was an award-winning artist specializing in watercolor painting. She received recognition as the poster artist for the Space Coast Art Festival and received awards in numerous shows and exhibitions in and around Brevard County, New Jersey, and the Rehoboth, Lewes, and Cape Henlopen, Delaware area. She was active in the Florida Watercolor Society, the Brevard Arts Council, and numerous other artist groups and galleries. Her skills live on in the students she taught in classes down the years and inherently through her children and grandchildren.

She was an active boater and sailor for many years, and was active with the Grumpy Ole Sailors and the East Coast Sailing Association with her husband, John. As a pair, they took numerous camping trips all around Florida, the United States, and internationally. Despite her desire to always run off and have the next adventure, she was the steadying influence in her household for her whole life. For many years she attended mass and was active in the church and organizations at Our Savior Catholic Church in Cocoa Beach, and was an amazing wife, mother, grandmother, great-grandmother, and friend.

She is survived by her three children, Kevin Schnitzius and his wife Kim, Danielle Schnitzius Norwood and her husband Charlie, and Mark Schnitzius and his wife Marjorie; her five grandchildren, Abi Schnitzius, Livia Schnitzius, Bryce Schnitzius, Chaeli Norwood Harden (Kevin), and Chase Norwood, and by her great grandson, Kix Harden. She is preceded in death by her husband, John Schnitzius, her parents, Felix and Helen, and by her brother, Tom Thompson.

Services will be held graveside at 2pm on Tuesday, March 19th at Florida Memorial Gardens in Rockledge. In lieu of flowers her family asks that donations be made in Loretta’s name to the Cancer Center of Merritt Island.

And my Facebook post about Mom.  

I owe her a longer post but I couldn't find the words at the time.  So this was all:

And now, Mom.  Just a month and two days after Dad.  We're sad, but they would have chosen to go even closer together if they could.

 

I'll leave Marjorie to decide whether and what she wants to post here about her mother.  I'm at least glad I got to be with her at the end, and to be there for Marjorie.  Those were some awful days.

Menorca

 

We've just returned from a 6 night trip in Menorca (Santo Tomas). The weather was lovely, and we experienced the best snorkelling we've done in Europe, but the trip was still only almost relaxing due to the pandemic. 

Starting from the beginning there were a number of additional stressors which I'll outline below (for the fun of reliving!) which took away from a fully relaxing holiday: 

1.   Mark had issues logging onto the NHS Covid Pass website, which said it needed to verify his identity the day before our trip (and this could take up to 7 days) even though he'd logged in the week prior with no issues. In the end he had his information confirmed in a few hours, but it was a stressful few hours. 
2. The train to the airport was sold out - this has never been an issue, and I didn't think to worry about it (silly me!). We booked a taxi to take us to the airport instead, and realised at pick up time that the booking was for the wrong day! Luckily the taxi company was able to send a cab within 10 minutes anyway, but again it was really stressful. I'm sure Mark provided the correct details when he made the booking. 
3. There was a car wreck (and delay) on the way to the airport. Luckily this didn't slow us down too much in the end, but was another 'oh shit' moment, and it felt like we were not going to be able to go on this trip. 
4. People are assholes! Nothing new here, but on full display at the airport via the number of people who didn't wear their masks correctly or at all. We were in FP2 masks and face shields and using copious hand gel in response, but feeling the need to be so vigilant was not relaxing.

I felt pretty frazzled by the time we got there and am happy to report that once we arrived to Menorca things went pretty smoothly. Menorca is beautiful and the weather was great. If we did it again, we'd stay somewhere different, where we could self-cater as the hotel buffet was pretty mediocre. But our room was nice and we had a gorgeous view of the sea. The hotel pool was pretty great, and we were right on the beach. Also, there were no children at the hotel which was a big perk. 

The snorkelling was the best we've experienced in Europe so far. I saw two eels, no other 'big ticket' animals, but plenty of fish and lots of varieties. The visibility was amazing, the water was clear at 30+feet, and the island off the shore (near Es Bruc- restaurant) was easy to circumnavigate and provided really excellent snorkelling conditions. 

Es Bruc was the best restaurant in the area by far, but we didn't have any meals on this trip that I'd call 'gourmet'. If we go again we'll rent a car as well as there may have been better options a bit further afield. 

On the whole I think Menorca was doing pretty well with Covid safety compliance. Most people were good about wearing their masks and we were mostly outside which made things feel much safer, but there were unavoidable covid stressors due to UK precautions before returning including a covid test (not fun, but understand the reasons) and a four page document to complete 2 days before returning. We did manage to let down our guards enough to book massages, which we've not done since covid and that was really nice. 

Ultimately I'm glad we were able to go and glad to have had a week of real summer (and the suntans to show for it). 

Hamish also seemed to have a nice time with the dog sitter which was a real plus. He apparently liked them so much he was sleeping with the youngest son of the family by the last night. Little bugger. 

Parkrant

Parkrun was my soccer replacement after blowing out my knee. A weekly timed 5K run that promised to be forever free, tracked your statistics for you, and was a worldwide phenomenon, it scratched a lot of my itches, and was this shiny awesome thing. Of course it had to shut down for the pandemic like every other mass gathering, but they soon worked out a system (NotParkrun) whereby you could just do your own socially-distanced run during the week on the honor system and they'd still track it for you.

A lot of the shine has gone off it for me though. Not the running -- I'm still doing NotParkruns more than twice a week on average. Rather, I'm not keen on how both Parkrun and my fellow parkrunners are handling the proposed restart.

It's partly a victim of its own popularity. I hadn't anticipated this happening at all, but for Parkrun to restart, a significant number of parkruns have to restart simultaneously. Why? Because if only a few restart, they will be flooded by parkrunners from neighboring runs that are shut down, creating unsafe conditions. So Parkrun had to get agreements from a lot of landowners, who have responded to the restart plan with varying degrees of enthusiam, for a variety of reasons. This is where Parkrun really seemed to screw up their response: unbelievably, they got pissy with the landowners and invalidated their concerns.

Worse, Parkrun's stance seems to have been restart, restart, restart. While they've been compliant with government mandates, I get the sense that they'd definitely restart if government said it was okay but the scientists said it wasn't. So I'll be having a careful look at things before I ever run one again.

They seem to have worked it out with enough of the landowners now, but a lot of the agreements were contingent on the government proceeding with the last stage of lockdown easing, and that's just been put off until 19 July. So Parkrun has made the announcement that they have to delay the restart, and now it's the other parkrunners that have irked me. They have the same restart, restart, restart mentality, and are claiming the delay has no basis in science (spoiler: it does). They're blaming the landowners as well. There's even a group that are proudly meeting at the usual parkrun time in our local park and running in defiance of the national organisation's request that people not do this. Jerks.

I'll still log my runs, and I'll go back (eventually) if they restart, but I'm really hoping things improve.

I can blog too!

I still wish we had kept blogging, as it still makes a much better record of our life than Facebook messages. I still use the blog as a memory assist way more than FB and their crappy search.

I didn't leave the village (even to go into Cambridge) for the whole month of February. We're doing our level best to keep our lives entertaining, but we've missed out on a year of exploring the UK and Europe now. I have a big London map that I stare at wistfully -- it's only an hour away and we haven't been in the past year. We're optimistic that things will improve, though. Marjorie's had her first vaccination, and will have her second next month. And me, I just the NHS text message on Friday, and will be getting my first shot on Thursday. That's the best news I've had in a while!

It's certainly been a memorable year, but because of the lack of variety, I'm worried that I'm going to remember it down the years as a single event, like you remember a concert. We stopped getting takeaway even, though we are trying something novel this month -- going meatless. My brother commented on Facebook that we would be surprised by how many options we'll have, and that's definitely been true. It hasn't been a struggle AT ALL. In fact, I've barely noticed that we're doing it, except for the novelty in dishes that Marjorie has been whipping up. I think it would be a lot harder if we were going out to eat more, and having to choose less tastier-sounding options on the menu. I don't think we'll be permanently meatless but it will certainly effect some sort of permanent change in how much meat we eat.

A year of Covid-19

A year ago today the WHO declared Covid-19 a global pandemic, we had just returned from Belgium and planned to go to Florida the following week for Mark's nephew's wedding but our travel plans were cancelled (obviously). At the time it was exciting but abstract, the actual threat didn't seem real, and I suppose we thought it would be like our SARs experience in Singapore and that life would largely go on as usual, and it would all blow over in a few months. 

Now, a year later I'm wondering if we'll every fully return to normal. We've been extremely fortunate throughout this, but it's still been hard. I'm so ready to get back into the world, but know it's likely to feel strange, and scary. I've had my first vaccination (Astra Zeneca) and have the 2nd scheduled for early April. Mark hasn't had his yet (hopefully he'll get a call this month), and we will feel much better 14 days post his first vaccination. I hope we'll be able to travel this summer! It's nice here, but it feels like a tease having the rest of Europe on our doorstep but not being able to go there. 

We've taken lock-down very seriously and haven't gone anywhere in months (including grocery stores - deliveries and click and collect almost exclusively). I miss restaurants/pubs, but, I've definitely become a better cook this year. I was an okay cook pre-pandemic, but as we haven't even had take away in months I've become more adventurous and tried more laborious recipes that I would have previously not felt I had time for. Can't use that excuse now :^)! This month we're having a 'meat-free' March, if only to do something different. It's going well, and I don't think either of us have missed meat as much as we thought we would (but still 20 days to go). Vegan would be hard though, don't think we're ready to give up eggs or milk anytime soon. 

In other news, I finish my current job in two weeks and start a new job in April. The new role is better paying and a permanent position, and hopefully a role I'll enjoy. It's taken 4 years to get back to a position and wage that feels more commiserate with my work experience - the real downside of having to start over again (at least partly) every time we move. 

Sick puppy

Hamish celebrated his ~11th birthday last Sunday. We spoiled him all week beforehand, making him better meals than usual (which is saying something because I cook for him anyway). On the actual day we took him on a long walk to Fen Ditton and gave him a lamb casserole dinner. He was smiley all day and seemed to be loving life. 

The next day however, he was very, very unwell. I woke up before 5am to the noise of Mark escorting Hamish outside after he'd already vomited repeatedly in the house. I was initially worried he might have an abdominal blockage because he ate an entire beef rib on Saturday (it was thinly cut, but I would normally not allow him to have the whole thing in one sitting and he ate the thing in less than 3 minutes so couldn't have chewed it much). I made an emergency vet appt for him and he spent the next 36 hours at the vet having blood tests and x-rays to try to figure out what was making him so unwell. Luckily he didn't have a blockage, and didn't require surgery, but his liver enzymes were elevated, and he was clearly very unwell. Best guess is that he ate something that made him unwell (he eats random crap when we go out on walkies so who knows what it was). The whole situation was incredibly upsetting and I spent most of Monday ugly crying in between work calls. 

He's home now and seems to be improving but he's still very picky about what he eats, which is not normal for our boy (he's a pig). The vet sent him home with sensitive stomach canned food, but he won't touch it so I've been making chicken rice for him. He's eating the chicken and keeping it down, which is reassuring. Day by day he seems to be improving and slowly getting back to his normal self. 

Long way to say it's been a stressful week, but things seem to be mostly okay now.