Friday, 27 April 2012

Pythagorean triples

A Pythagorean triple (a,b,c) consists of positive integers a, b and c such that

a^2 + b^2 = c^2.\

The triple is called primitive if a, b and c are relatively prime.  It can be easily proved that all Pythagorean triples can be generated using Euclid's formula:

 a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2

where m and n are positive integers with m > n.  The triple (a,b,c) generated is primitive if m and n are coprime and m - n is odd.  There are 16 primitive Pythagorean triples with c < 100:

( 3 , 4 , 5 )( 5, 12, 13)( 7, 24, 25)( 8, 15, 17)
( 9, 40, 41)(11, 60, 61)(12, 35, 37)(13, 84, 85)
(16, 63, 65)(20, 21, 29)(28, 45, 53)(33, 56, 65)
(36, 77, 85)(39, 80, 89)(48, 55, 73)(65, 72, 97)


There are a lot of interesting facts about primitive Pythagorean triples.  For example:
  • One of a and b is odd, the other is even; c is always odd.
  • Exactly one of a and b is divisible by 3.
  • Exactly one of a and b is divisible by 4.
  • Exactly one of a, b and c is divisible by 5.
  • Every prime factor of c leaves a remainder of 1 when divided by 4.

A generalization of the Pythagorean triples leads to the famous Fermat's Last Theorem dated 1637, which remained an unsolved mystery for 358 years until finally proved by Andrew Wiles in 1995.  The theorem states that the equation

a^n + b^n=c^n \!

has no non-trivial solutions for (a,b,c) if n is a positive integer greater than 2.

Thursday, 26 April 2012

One Billion Years (Past And Future)

Scrap Metal Gallery's current group show Read All Over explores the state of language in books and visual art.  All the works on view incorporate language and/or books, confronting the question "Is visual art a refuge for words, a place where text can achieve some permanence, while words have become mere ephemera in this world?"

Dave Dyment's One Billion Years (Past And Future) at Scrap Metal, Toronto 2012:


Wednesday, 25 April 2012

Saturday, 21 April 2012

Curved Harris

Lawren Harris reinterpreted on the curved façade of the AGO (Art Gallery of Ontario), Toronto 2012:

Friday, 20 April 2012

Et Cetera

Hoxton Art Gallery's current group show Et Cetera explores the notion that meaning and significance can be found in things that are commonly deemed uninteresting or unworthy. Although I won't have a chance to see this exhibition in London, the idea of focusing on various overlooked elements seems rather refreshing in this increasingly repetitive urban life. The show is running until May 24.

Laura Culham is one of the artists showing at Et Cetera:


Image courtesy of Hoxton Art Gallery

Wednesday, 18 April 2012

Verburg's Untitled

(I see your point, and I understand your perspective - maybe there is no clear direction, no right or wrong answer.  Maybe I'm just seeing it differently - or maybe we're saying the same thing)

Jim Verburg at Convenience Gallery, Toronto 2012:


A construction site in downtown Toronto, 2012:

Tuesday, 17 April 2012

Blue Moon

Looks like 2012 has been a busy year for Jürgen Paas so far. In addition to his current appearances at Art Cologne and Ladenburg's Galerie Linde Hollinger, which I have previously written about, his solo exhibition Blue Moon at Cor Galerie is concurrently running until May 5.

Paas at Cor, Zürich 2012:


Image courtesy of Jürgen Paas

Monday, 16 April 2012

Exit, Enter

It is hard for the eyes to stay away from Celia Neubauer's canvases.  There is always so much to see in her contrasting layers of landscape in each of her recent paintings, which are not very different from collages in form and spirit, with an obvious influence of traditional oriental painting.  Her solo exhibition Exit, Enter is currently on view at General Hardware Contemporary.

Neubauer at General Hardware, Toronto 2012:





Neubauer's smaller-scale monochromatic sketches are equally enticing:



General Hardware is quickly becoming one of my favourite galleries in Toronto.  Thanks, N, for your hospitality and for another great show.

Sunday, 15 April 2012

Road Shots

At the current exhibition Road Shots by Public Studio (Elle Flanders & Tamira Sawatzky) are a series of photographs tracing the political landscape at the core of the Israeli occupation of Palestine.  The photographs were superimposed with patterns of Islamic architectural ornamentation cut-outs.  Geometry in Islamic architecture is a way to order the chaotic world.

Public Studio at O'Born Contemporary, Toronto 2012:


Saturday, 14 April 2012

Köln Ladenburg Paas

Jürgen Paas's works currently on view at two latest shows ...

At Art Cologne for Salon Verlag, April 18-22:


At Galerie Linde Hollinger, Ladenburg, till May 24:


How I wish I could fly to Germany now!

All images courtesy of Jürgen Paas

Friday, 13 April 2012

Abstract/architectural Milner

Nancy Milner's architecturally inspired works at Premiums, interim projects by postgraduate students of Royal Academy Schools, presented a tension between representation and abstraction.  Often using her own photographs as a starting point, Milner created works that can both be viewed as abstraction or as architectural images.

Milner at Royal Academy of Arts, London 2012: