Friday, June 26


Spin is my first acquaintance with a Jos Bergmans creation, realized here in exotic woods by Brian Menold. Bergmans' polycube constructions are for me a natural progression from my old standby, Stewart Coffin's Convolution, and its subsequent improvement, Involute. Spin is a pleasure to take apart and put back together.

Monday, June 15

Japanese tree lilac

At this time of the year, my walks to and from the park are distracted by these two mature Japanese tree lilacs — Syringa reticulata — in bloom. The pervasive fragrance of the showy flowers is to me quite pleasant, an opinion not apparently shared by everyone. One person described it as vaguely fishy but — as another commenter there pointed out — it is musky, not muskie.

Sunday, June 14

Free ride

Today was Metrolinx's appreciation day for the Bloor and Weston communities putting up with the rail service construction's inconveniences. In spite of a steady rain, the event seemed well attended. I picked up my free round-trip ticket and went up to the downtown platform where I watched a full up train come and leave without me. I then proceeded to the airport platform for what turned out to be more of the same. One might have expected a small ramp up of services (considering the freebie) but I guess appreciation only goes so far. Standing in a crowded passenger vehicle for an hour is not my idea of a worthwhile venture. I left the platform, availed myself of a free hot dog, and sauntered home.

Sunday, June 7

Weston Station

A couple more pictures of the new Union Pearson Express train. Above, a train in Weston Station — headed downtown (the top of the CN tower is visible above the front of the train). The station is far from ready. I had to dodge workers to get to the platform — and the new footbridge across Lawrence Avenue West was still closed! Below, a train has left the station going to the airport — about to enter the tunnel that allows Weston's King and Church streets not to be interrupted by the UP service. John Street was a third major Weston thoroughfare (roughly where the train is in the photo). It being too close to the Lawrence rail overpass, the railroad section of John had to be sacrificed to the tunnel approach's downslope. In lieu of the road, residents have been promised a footbridge.

Saturday, June 6

UP and away

The Union Pearson Express trains started their passengers-accepted runs this morning. My first photo shows a three-parter shortly after it had left Weston station (one of only two stops: a 7-minute walk from my home) on its 25-minute ride from the airport to downtown Toronto. The outrageously high fares are mitigated somewhat if one buys a transit debit card for one's payments. My second photo, a few minutes later, shows a two-parter from Toronto about to arrive into Weston station:

Tuesday, May 26

Indexing the Leyland primes

A Leyland number is x^y+y^x, technically x >= y > 1 except that (x,y) = (2,1) has lately been admitted. Ordered by size, every Leyland number has associated with it an index number. Leyland primes are Leyland numbers that are prime, proven or probable. Most of the known Leyland primes can also be indexed — but not all of them! The difficulty lies in the manner in which Leyland primes have historically been discovered: restricting y to small values while allowing x to be very large. This method divorces finds from their Leyland number indices (i.e., their size), with the possibility of unchecked primes existing between two known examples. Of the currently 1091 known Leyland primes, I believe that only the smallest 954 (give or take) are indexable.

There's a Numberphile video on Leyland numbers and Leyland primes. In it is mentioned the currently largest known Leyland prime: Serge Batalov's (x,y) = (328574,15). What might be this 386434-digit prime's index? Step one is to figure out the number's Leyland index. Using a Mathematica program to count, I believe it to be Leyland #3808683611. Step two is to fit the Leyland number indices of the 954 indexable primes to a curve:

The suggestion here is 17*index^2.23 as a decent fit. This equation is not meant to be exact: a database of further-along primes might necessitate adjusting the multiplier and exponent somewhat, but for our purposes it is good enough. What Leyland prime index will generate a Leyland number index of ~3808000000? The number 5553 comes close. So, I expect the currently largest known Leyland prime to be roughly #5550 of all Leyland primes. That leaves thousands of smaller Leyland primes still to be discovered!

Saturday, May 16


Year after year, I see here a lot of the same bird species (and, I rather suspect, the same individuals where those species are poorly represented). I first saw this trumpeter swan in the river on May 7, and again the following day. I thought it had moved on, but there it was yesterday — flying a couple of circles before heading upriver. I know there are trumpeters on the lake (Ontario) but this is the first one I've seen in this locale. Ontario trumpeters sport yellow wing-tags for easy identification.

Thursday, April 16

O Canada

The image is a section of Canada's northwest, part of a new political map of Canada, a copy of which I have placed here. It's a big picture, so be patient. For me, it loads and handles ok in Chrome and Firefox — not so much in Safari. For those who didn't know, I'll point out that all of the islands in Hudson Bay — including its southerly James Bay extension — are part of Nunavut.

Wednesday, April 1

My hood

The picture is an Apple 3D representation of where in Toronto I live and take my walks. I'll try to describe the map in words without recourse to symbols planted onto the image. The bird's eye view looks west toward Raymore Park on the farther side of the Humber river, scene of significant devastation when hurricane Hazel hit in 1954. A built-in-1995 footbridge across the river is visible at the top. The main street is Weston Road, seen curving at the right toward the highrises. Just beyond is downtown Weston — once a village (the orange lot near the bottom right of this 1878 map is my reckoned property), then town, outside of Toronto. The railroad corridor shown under reconstruction at bottom right will hopefully be finished this year.

If you look for an h — let's call it a chair — with legs abutting Weston Road, my street — Sykes Avenue — is the seat and front leg of that chair. My residence is on the south side of the leg part, sixth house in. Sykes runs into Denison Road West, the back of the chair, which then curves and continues until it is stopped by St. John's Cemetery on the Humber, a private cemetery that — strictly speaking — is in Mount Dennis, the community south of Weston. To the right (north) of the cemetery is Denison Park. It is behind this park — looking down — that I get my photos of Raymore island (as I call it) sitting in the deeper water held back by the curved whitewater of Raymore weir, downstream to its left (south).

The cemetery and most of the streetscape is on high ground, contrasted with the low ground of Raymore Park, adjacent bits on the nearer side of the river including the autumn-hued treescape above (west of) the cemetery, and the large school building and mega-housing structures at bottom left, below (east of) the cemetery.

A photo I took this morning of a beaver in the river returning to its den under the island:

Wednesday, March 11

Visitors from infinity, reprise

When I posted my Visitors from infinity on New Year's day, I had been working with a database of two-and-a-half billion (10^9) numerical correspondences. On February 6, I completed a computation taking the Yellowstone permutation to five billion — but only as another two-and-a-half-billion-term file, because I cannot (with only 64 GB RAM) store all five billion terms into Mathematica at the same time.

The limitation of that shortcoming is that when I map n into A098550(n) or A098550(n) into n, any time the mapping crosses over into the other-file regime I have to reboot Mathematica with that other file. Just the reading of it takes about nine hours and working the 302 currently unknown-outcome trajectories (with minima < 10^4) backwards (towards the left in the graph) might take another day or two on each reading. But I finally completed the task yesterday! The picture shows all 302 orbits with their minima synced (although many of the more-or-less random-hued paths are obscured by others).

One can see in the graph that my backward reach is limited to five billion, while moving forward (to the right) always ends in a point beyond (sometimes well beyond) that. If you are interested in the raw data, it is still here — and I have placed individual graphs for all 302 chains here.