The image above is based on the same formulas as the image below. The one above is just the upper left 16 by 16 cells.
Sunday, March 30, 2008
GCD Plots on a Curve
My latest project is to create studies for color grid paintings, based on the coprimes and greatest common divisor (GCD) digital drawings, but to make them curved. Here's the initial attempt.
The image above is based on the same formulas as the image below. The one above is just the upper left 16 by 16 cells.
The image above is based on the same formulas as the image below. The one above is just the upper left 16 by 16 cells.
Thursday, March 20, 2008
Friday, March 14, 2008
Ten GCD Plots
Here are ten plots, with the 47X47 column and row numbers figuring in a pair of formulas, which then are tested for a greatest common divisor, GCD, of one (coprimes), two, and three. The background is yellow. If the GCD is one, two, or three, the cell is black, blue or red, respectively. The only things that vary are the formulas. For each plot there are two formulas, with the formula variables being row number and column number. For each cell, the two formulas are calculated, then the GCD of the numerical results are tested for one, two, or three. The cell color is set accordingly. If the GCD is something other than one, two, or three, the cell remains yellow.
a = row number
b = column number
If b>a then a and b are swapped. The formula pairs are:
1. a, b
2. a3+b3, a3–b3 (Taxicab and cabtaxi candidates)
3. b2+b+1, a2+a+1
4. b+a, b–a
5. b2+b+2, a2+a+2
6. b2–b+2, a2-a+1
7. b2+1, a2+2
8. b2+2b*a+a2, b2–2b*a–a2
9. b3+b2+b, a2+a
10. 3b3+2b2+b, 2a2+a
a = row number
b = column number
If b>a then a and b are swapped. The formula pairs are:
1. a, b
2. a3+b3, a3–b3 (Taxicab and cabtaxi candidates)
3. b2+b+1, a2+a+1
4. b+a, b–a
5. b2+b+2, a2+a+2
6. b2–b+2, a2-a+1
7. b2+1, a2+2
8. b2+2b*a+a2, b2–2b*a–a2
9. b3+b2+b, a2+a
10. 3b3+2b2+b, 2a2+a
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Monday, March 10, 2008
Cubes, the Cabtaxi Number Fleet
A taxicab number is the smallest integer that is the sum of two positive cubes from n different sums. So there's one taxicab number for n = 1, 2, 3. . . A cabtaxi number is the smallest positive integer that is the sum of two positive or negative integers from n different sums. (See Wikipedia or Wolfram.) Only nine cabtaxi numbers are known, though more are supposed to exist even though they haven't been found yet. All but the first three are seven digits or more. It occurs to me that the numbers for real taxicabs should be just a few digits — more than one, but less than seven certainly. So I think it would be interesting to find all the two, three, or four digit candidates for cabtaxi numbers (the cabtaxi fleet) , and it shouldn't matter whether they're the smallest integer for n sums.
Therefore, a cabtaxi candidate is like a cabtaxi number except that it may not be the smallest integer; and, for this project I'll say that it should be from 2 to 4 digits. This set of cabtaxi candidates would include Cabtaxi(2) and Cabtaxi(3) — 91, 728 — as well as the taxicab number, 1729, which is Taxicab(2).
Here are a few of the candidates in the cabtaxi fleet, but probably not all:
91 = 63 – 53 = 33 + 43, Cabtaxi(2)
152 = 63 – 43 = 33 + 53
189 = 63 – 33 = 43 + 53
217 = 93 – 83 = 63 + 13
513 = 93 – 63 = 83 + 13
721 = 93 – 23 = 163 – 153
728 = 93 – 13 = 123 – 103 = 63 + 83, Cabtaxi(3)
1027 = 193 – 183 = 103 + 33
1729 = 13 + 123 = 93 + 103, Taxicab(2)
1736 = 183 – 163 = 123 + 23
3367 = 163 – 93 = 343 – 333
4104 = 183 – 123 = 153 + 93 = 163 + 23
5824 = 183 – 23 = 243 – 203
5859 = 193 – 103 = 273 – 243
7922 = 203 – 23 = 243 – 183
8216 = 383 – 363 = 203 + 63
8587 = 543 – 533 = 193 + 123
From this we should be able to develop a way to use cabtaxi candidates graphically, similar to the coprimes plots.
Two ways to look at Cabtaxi(2):
Therefore, a cabtaxi candidate is like a cabtaxi number except that it may not be the smallest integer; and, for this project I'll say that it should be from 2 to 4 digits. This set of cabtaxi candidates would include Cabtaxi(2) and Cabtaxi(3) — 91, 728 — as well as the taxicab number, 1729, which is Taxicab(2).
Here are a few of the candidates in the cabtaxi fleet, but probably not all:
91 = 63 – 53 = 33 + 43, Cabtaxi(2)
152 = 63 – 43 = 33 + 53
189 = 63 – 33 = 43 + 53
217 = 93 – 83 = 63 + 13
513 = 93 – 63 = 83 + 13
721 = 93 – 23 = 163 – 153
728 = 93 – 13 = 123 – 103 = 63 + 83, Cabtaxi(3)
1027 = 193 – 183 = 103 + 33
1729 = 13 + 123 = 93 + 103, Taxicab(2)
1736 = 183 – 163 = 123 + 23
3367 = 163 – 93 = 343 – 333
4104 = 183 – 123 = 153 + 93 = 163 + 23
5824 = 183 – 23 = 243 – 203
5859 = 193 – 103 = 273 – 243
7922 = 203 – 23 = 243 – 183
8216 = 383 – 363 = 203 + 63
8587 = 543 – 533 = 193 + 123
From this we should be able to develop a way to use cabtaxi candidates graphically, similar to the coprimes plots.
Two ways to look at Cabtaxi(2):
Sunday, March 9, 2008
Friday, March 7, 2008
Wednesday, March 5, 2008
Fragments of the world
Nothing new, but it bears repeating.
A system: Sol Lewitt, 2003. Interview by Saul Ostrow, in Bomb Magazine, Issue 85. LeWitt: "What [the wall drawings] looked like wasn’t important. It didn’t matter what you did as long as the lines were distributed randomly throughout the area. In many of the wall pieces there is very little latitude for the draftsman or draftswoman to make changes, but it is evident anyway, visually, that different people make different works. I have done other pieces that give the draftsperson a great liberty in interpreting an action. In this way the appearance of the work is secondary to the idea of the work, which makes the idea of primary importance. The system is the work of art; the visual work of art is the proof of the system. The visual aspect can’t be understood without understanding the system. It isn’t what it looks like but what it is that is of basic importance."
Selected Image
A process: Richard Serra, 2006. Interview with Phong Bui, in The Brooklyn Rail, June, 2006. Serra: "There are certain conditions that are a given and that you can rely on. In sculpture gravity is undeniable. Sculptural form must necessarily confront gravity. I am interested in process and matter, in construction, in how to open up the field. The problem for me is to address within a work circulation or movement that is outside of all representation; that is to make movement itself the subject which generates or constitutes the work. My development has been up to this point fairly logical and sequential. But it’s crucial for me to pay attention to how the work develops and maintain a critical and fresh dialogue with what it is that I’m doing and what I’m intending to do, and then try, if I can, to make the most radical breaks each time out, however, it doesn’t always happen that way."
Selected Image
Fragments of the real world: Ellsworth Kelly, 2006. Touching the Void, by Gordon Burn, The Guardian, guardian.co.uk. Burn: "There is a painting in the Serpentine show that is based on colours from the less doomy part of Rothko's palette: a large rectangle of orange is combined with smaller rectangles of pale green and yellow. The crucial point for Kelly - it is the one he established his reputation making - is that the three colours are contained within their own discrete panels and not painted on to a single canvas. The hard edges that separate them preserve the integrity of each colour. 'There's no dominance here,' Kelly says. 'It's like the relationship between the two of us: I'm a body, you're a body. If I did it as a single painting, the orange would be the main colour and the others its satellites. By doing it in panels, each has its own uniqueness. It's the difference between depicted space - a painted image - and literal space. I feel my paintings are fragments of the world and I'm simply digging them up and presenting them. I want to get more into the real world.' "
Selected Image
A system: Sol Lewitt, 2003. Interview by Saul Ostrow, in Bomb Magazine, Issue 85. LeWitt: "What [the wall drawings] looked like wasn’t important. It didn’t matter what you did as long as the lines were distributed randomly throughout the area. In many of the wall pieces there is very little latitude for the draftsman or draftswoman to make changes, but it is evident anyway, visually, that different people make different works. I have done other pieces that give the draftsperson a great liberty in interpreting an action. In this way the appearance of the work is secondary to the idea of the work, which makes the idea of primary importance. The system is the work of art; the visual work of art is the proof of the system. The visual aspect can’t be understood without understanding the system. It isn’t what it looks like but what it is that is of basic importance."
Selected Image
A process: Richard Serra, 2006. Interview with Phong Bui, in The Brooklyn Rail, June, 2006. Serra: "There are certain conditions that are a given and that you can rely on. In sculpture gravity is undeniable. Sculptural form must necessarily confront gravity. I am interested in process and matter, in construction, in how to open up the field. The problem for me is to address within a work circulation or movement that is outside of all representation; that is to make movement itself the subject which generates or constitutes the work. My development has been up to this point fairly logical and sequential. But it’s crucial for me to pay attention to how the work develops and maintain a critical and fresh dialogue with what it is that I’m doing and what I’m intending to do, and then try, if I can, to make the most radical breaks each time out, however, it doesn’t always happen that way."
Selected Image
Fragments of the real world: Ellsworth Kelly, 2006. Touching the Void, by Gordon Burn, The Guardian, guardian.co.uk. Burn: "There is a painting in the Serpentine show that is based on colours from the less doomy part of Rothko's palette: a large rectangle of orange is combined with smaller rectangles of pale green and yellow. The crucial point for Kelly - it is the one he established his reputation making - is that the three colours are contained within their own discrete panels and not painted on to a single canvas. The hard edges that separate them preserve the integrity of each colour. 'There's no dominance here,' Kelly says. 'It's like the relationship between the two of us: I'm a body, you're a body. If I did it as a single painting, the orange would be the main colour and the others its satellites. By doing it in panels, each has its own uniqueness. It's the difference between depicted space - a painted image - and literal space. I feel my paintings are fragments of the world and I'm simply digging them up and presenting them. I want to get more into the real world.' "
Selected Image
Saturday, March 1, 2008
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