Monday, July 13

It's bigger than we thought

Phil Plait's Bad Astronomy blog (for Slate) this morning had me in a wee bit of a tizzy. One might have expected better. Plait's contention that the very preliminary New Horizons Pluto diameter of 2370 ±20 km somehow betters the previous estimate — which he thought was 2368 ±20 km — is just silly and (likely because a few of his readers said so) he has since done a couple of updates to soften the headline.

It gets worse. It is true that there was — and continues to be — a lot of confusion about Pluto's diameter. I don't know who first thought to add the ±20 to the 2368 km but a number of current online planetary sources had it. However, the 2368 almost certainly comes from an article about the variation of methane abundance in Pluto's atmosphere. The authors suggest:

For the sake (and amusement) of making predictions, we "guess" here a radius of 1184 km, based on our considerations on its likely lower (1180 km) and upper (1188 km) limits.

So their estimate is actually 2368 ±8 km! In his second update Plait says: "Emily Lakdawalla in the article linked in the first update says that the new measurements reduce the uncertainty quite a bit, and I'm inclined to agree." Really? Why would Lakdawalla suggest this? Last year Emily Lakdawalla authored an article titled When will we know which is bigger, Pluto or Eris? In it, Lakdawalla quotes Alan Stern (principal investigator of the New Horizons mission) saying that "we already know the radius to ± 20 kilometers", to which she explains (in brackets) that "they know the diameter to ± 40 kilometers". Wow!

Wednesday, July 8

Strength in numbers

For many years now I have been an admirer of the mechanics of swallowwort growth. If two or more of the spindly stalks are sufficiently close together, they will (on contact) braid themselves into a more sturdy (rope-like) superstalk. The additional strength of this structure allows the plants to reach higher, before weight (eventually) arches them over. If under a tree, that extra push may be all that is needed to reach a lower branch. The vine (Cynanchum rossicum) is invasive in North America and there is plenty of it in the local (private) cemetery. A relative of American milkweed, the thickets created by the plant have given it the moniker dog-strangling vine.

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 1092 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.