Friday, 30 June 2017

17

It's a well known fact that the proper fractions with denominator 7 are recurring decimals which share the same digits appearing cyclically:

1/7 = 0.142857
2/7 = 0.285714
3/7 = 0.428571
4/7 = 0.571428
5/7 = 0.714285
6/7 = 0.857142

It turns out that the 7ths are not the only fractions whose decimals exhibit this cyclic property.  So are the 17ths:

1/17 = 0.0588235294117647
2/17 = 0.1176470588235294
3/17 = 0.1764705882352941
4/17 = 0.2352941176470588
5/17 = 0.2941176470588235
6/17 = 0.3529411764705882
7/17 = 0.4117647058823529
8/17 = 0.4705882352941176
9/17 = 0.5294117647058823
10/17 = 0.5882352941176470
11/17 = 0.6470588235294117
12/17 = 0.7058823529411764
13/17 = 0.7647058823529411
14/17 = 0.8235294117647058
15/17 = 0.8823529411764705
16/17 = 0.9411764705882352

In fact, there are infinitely many of them: 7, 17, 19, 23, 29, 47, 59, 61, 97, ...  You must have noticed that they are all prime numbers.  They are called full reptend primes (which has 10 as their primitive roots, for those of you, like me, who are interested in algebraic number theory).

Thursday, 29 June 2017

Jenkins surrealism

Couldn't say I love this show but his sculptures, which recall René Magritte and other art historical imagery, make the show quite interesting.  Mark Jenkins currently at Arsenal Toronto:





Tuesday, 27 June 2017

Saturday, 24 June 2017

Full Spectrum + blended colours

Something for Toronto's Pride weekend ...

Public Studio's public art installation Full Spectrum at Maitland and Church Streets is a commission for WorldPride Toronto, 2014:





Found blended colours in uptown Toronto, 2015:


Friday, 23 June 2017

Thursday, 22 June 2017

Irrational ^ Irrational

I came upon this very clever but simple proof recently ...

It is not known which irrational numbers x and y would make xy rational.  But it can be easily proved that such rational xy exists.  We know that 2 is irrational.  Take, for instance, the number


which must either be rational or irrational.  If it is rational, then we're done.  If it is not, then the number


must be rational.  So either case proves the existence of rational xy where both x and y are irrational.

Monday, 19 June 2017

Architectural compositions 2

More abstract compositions of architectural fragments ... all pictures were taken in London, 2016:





Sunday, 18 June 2017

Architectural compositions

I like using architectural fragments to create abstract compositions.  All pictures were taken in Toronto, 2017:





Saturday, 17 June 2017

Monday Night

Goto80's Monday Night from his latest album Floptrik just released yesterday:


Video from CPU Records' youtube channel

Friday, 16 June 2017

Wednesday, 14 June 2017

Size Matters

Their Size Matters series is currently showing at McMichael Canadian Art Collection.  I would love to see the show but it's a bit far; so I'd have to drive.  A sneak peek of Finn O'Hara and Steve Driscoll now at Angell Gallery, Toronto: