Need some math help!

[popinfresh]

OK, so I'm creating an artwork at the moment that is based off symmetry, math, Fibonacci's etc.
In the background is going to be radial lines coming from the centre of the painting to the edges, and I want the space between every one to gradually increase in proportion each time, coming from the centre towards the bottom of the canvas.

Is there an equation for something like this to work out the increasing angles? Or, a way to mark the edge of the canvas at certain points that steadily increase in distance towards the bottom?

The canvas is 40" high/20" from centre to bottom, and I would like about 15-16 lines including the lines dividing the canvas into 4 quarters.

If that doesn't make sense, I made this a little image that is attached, just mirror the bottom right corner to the left and that's the basic idea of lines that I need laid out. The top half I am just doing by eye.

Any help would be hugely appreciated!

[iamlowsound]

Just take the spacing and multiply it by a percentage, say 120%. Then take that spacing and do the same. Or you could just lay your bass next to it and mark at each fret.

lowsound

[mrphattay]

Aren't you talking about Linear Perspective -
A Vantage Point with Radiating Convergent Lines (orthagonals) at a fixed Horizon's Vanishing Point.
popular ie- Railroad tracks fading to the horizon.

This is a concept that has been known scientifically, and explored artistically, since the Renaissance.
There are many Art books & Papers on the subject.

rudimentary research on the net...an easy find.
(This took minutes...there are hundreds more)
ie- http://www.mos.org/sln/leonardo/expl...rspective.html
http://mathforum.org/sum95/math_and/.../perspect.html
http://studiochalkboard.evansville.edu/lp-intro.html
http://www.dummies.com/how-to/conten...rspective.html

But you know all this...since you mention Fibonacci (ie-his golden ratio)
So, I must be missing something.

[popinfresh]

Originally Posted by iamlowsound

Just take the spacing and multiply it by a percentage, say 120%. Then take that spacing and do the same. Or you could just lay your bass next to it and mark at each fret.

lowsound

Bass fret spacing is too small, however multiplying percentage makes sense. The only problem is working out the exact amount to multiply (or divide) so that the amount of lines will eventually line up directly with the quadrant lines. So, for example, if I am using 120% to multiply, beginning on the horizontal line, by the time I get to placing the last line before the vertical divide, I would like that to be a 120% increase in spacing as well.
I feel there's a real simple answer to this, using the amount of lines I will draw, but I'm just over thinking..

[popinfresh]

Originally Posted by mrphattay

Aren't you talking about Linear Perspective -
A Vantage Point with Radiating Convergent Lines (orthagonals) at a fixed Horizon's Vanishing Point.
popular ie- Railroad tracks fading to the horizon.

This is a concept that has been known scientifically, and explored artistically, since the Renaissance.
There are many Art books & Papers on the subject.

rudimentary research on the net...an easy find.
(This took minutes...there are hundreds more)
ie- http://www.mos.org/sln/leonardo/expl...rspective.html
http://mathforum.org/sum95/math_and/.../perspect.html
http://studiochalkboard.evansville.edu/lp-intro.html
http://www.dummies.com/how-to/conten...rspective.html

But you know all this...since you mention Fibonacci (ie-his golden ratio)
So, I must be missing something.

Definitely talking about linear perspective, however none of those allow me to find the correct spacing. Thanks though!

[Frank Tuesday]

Originally Posted by popinfresh

OK, so I'm creating an artwork at the moment that is based off symmetry, math, Fibonacci's etc.
In the background is going to be radial lines coming from the centre of the painting to the edges, and I want the space between every one to gradually increase in proportion each time, coming from the centre towards the bottom of the canvas.

Is there an equation for something like this to work out the increasing angles? Or, a way to mark the edge of the canvas at certain points that steadily increase in distance towards the bottom?

The canvas is 40" high/20" from centre to bottom, and I would like about 15-16 lines including the lines dividing the canvas into 4 quarters.

If that doesn't make sense, I made this a little image that is attached, just mirror the bottom right corner to the left and that's the basic idea of lines that I need laid out. The top half I am just doing by eye.

Any help would be hugely appreciated!

There are many different equations you could use depending on the type of growth you are looking for, linear, power, exponential, growth based on fibonacci numbers.

for example (7 divisions, the angles between lines are a,b,...,g)

fibonacci: a=30/11, b=30/11, c=60/11, d=90/11, e=150/11, f=240/11, g=390/11 (top)

90 = 7a+21n. Select either a or n (in degrees), and calculate the other. b=a+n, c=b+n... (middle, n=3, a=2.727)

90=a(1+x+x²+...+x⁶). b=a*x, c=b*x=a*x^2, d=a*x^3... (bottom, a=3, x=1.476)

[popinfresh]

Originally Posted by Frank Tuesday

There are many different equations you could use depending on the type of growth you are looking for, linear, power, exponential, growth based on fibonacci numbers.

for example (7 divisions, the angles between lines are a,b,...,g)

fibonacci: a=30/11, b=30/11, c=60/11, d=90/11, e=150/11, f=240/11, g=390/11 (top)

90 = 7a+21n. Select either a or n (in degrees), and calculate the other. b=a+n, c=b+n... (middle, n=3, a=2.727)

90=a(1+x+x²+...+x⁶). b=a*x, c=b*x=a*x^2, d=a*x^3... (bottom, a=3, x=1.476)

This is awesome, thank you

I like the growth achieved by the top and bottom the most.
I should have mentioned I haven't done above high school math in a long time, and never in school. I can understand the fibonacci sequence though one question, and sorry if missing anything, why is it divided by 11?

The second equation is somewhat there for me. Why is there a 21 to begin with? How do you decide what a and n are? I'm assuming 90 for the 90 degrees, 7 for the divisions obviously.

The third equation I'm completely lost. But if can explain it I'd love to figure it out!

[thefruitfarmer]

Originally Posted by popinfresh

I feel there's a real simple answer to this.....

The answer is 42.

[Frank Tuesday]

Originally Posted by popinfresh

This is awesome, thank you

I like the growth achieved by the top and bottom the most.
I should have mentioned I haven't done above high school math in a long time, and never in school. I can understand the fibonacci sequence though one question, and sorry if missing anything, why is it divided by 11?

The second equation is somewhat there for me. Why is there a 21 to begin with? How do you decide what a and n are? I'm assuming 90 for the 90 degrees, 7 for the divisions obviously.

The third equation I'm completely lost. But if can explain it I'd love to figure it out!

Answers:

fibonacci: you have a total of 7 angles, so using the first seven fibonacci numbers (1,1,2,3,5,8,13) we'll divide the 90 degrees into angles that correlate to the fibonacci numbers. So the last angle is 13 times wider than the first. The next to last angle is 8 times the first. So the sum of the first 7 fibonacci's is 33. That means that the first is 1/33 of 90 degrees which is 90/33 or 30/11ths. the last is 13 times wider, or 390/11ths.

2nds option: each angle grows by 'n' degrees over the previous angle. The first angle is 'a' degrees. so:
a = a
b = a+n
c = b+n=a+2n
d =c+n=a+3n
e = d+n=a+4n
f = e+n=a+5n
g = f+n=a+6n
90 = a+b+c+d+e+f+g = 7a+21n
You can choose 'a' or 'n' and calclulate the other, but 'a' has to be less than 14, and 'n' less than 4.8 (approximately).

The last is exponential growth. The first angle is 'a', and the rate of growth is 'x'.
a=a
b=ax
c=bx=(ax)x=ax²
d=cx=ax³
e=dx=ax⁴
f=ex=ax⁵
g=fx=ax⁶

90=a+b+c+d+e+f+g=a+ax+ax²+ax³+...+ax⁶.

[popinfresh]

Originally Posted by Frank Tuesday

Answers:

fibonacci: you have a total of 7 angles, so using the first seven fibonacci numbers (1,1,2,3,5,8,13) we'll divide the 90 degrees into angles that correlate to the fibonacci numbers. So the last angle is 13 times wider than the first. The next to last angle is 8 times the first. So the sum of the first 7 fibonacci's is 33. That means that the first is 1/33 of 90 degrees which is 90/33 or 30/11ths. the last is 13 times wider, or 390/11ths.

Perfect, thank you! One last question, how does 90/33 become 30/11ths? I'm understanding how the divisions are increasing easily, just not how to work out the 11ths part?

Originally Posted by Frank Tuesday

2nds option: each angle grows by 'n' degrees over the previous angle. The first angle is 'a' degrees. so:
a = a
b = a+n
c = b+n=a+2n
d =c+n=a+3n
e = d+n=a+4n
f = e+n=a+5n
g = f+n=a+6n
90 = a+b+c+d+e+f+g = 7a+21n
You can choose 'a' or 'n' and calclulate the other, but 'a' has to be less than 14, and 'n' less than 4.8 (approximately).

Gotchya! How did you figure out that 'a' had to be less than 14 and 'n' less than 4.8? Also, I'm still a little confused by the 21, but assuming the sum comes to 21 because b=a+n actually holds a numerical one before the 'n', correct?
Does this mean this could work with any numbers for 'a' and 'n', as long as they are below the previously mentioned values? Will it simply produce different angles? I don't have a protractor just yet to play around with this and make it visual which always helps me out.

Originally Posted by Frank Tuesday

The last is exponential growth. The first angle is 'a', and the rate of growth is 'x'.
a=a
b=ax
c=bx=(ax)x=ax²
d=cx=ax³
e=dx=ax⁴
f=ex=ax⁵
g=fx=ax⁶

90=a+b+c+d+e+f+g=a+ax+ax²+ax³+...+ax⁶.

I actually have no idea what sup stands for, but am going to look it up!

I didn't realise I was this rusty with math haha, it's fun getting back into it though!

[Frank Tuesday]

Originally Posted by popinfresh

Perfect, thank you! One last question, how does 90/33 become 30/11ths? I'm understanding how the divisions are increasing easily, just not how to work out the 11ths part?

Divide the numerator and denominator both (top part of fraction) by 3,

Gotchya! How did you figure out that 'a' had to be less than 14 and 'n' less than 4.8? Also, I'm still a little confused by the 21, but assuming the sum comes to 21 because b=a+n actually holds a numerical one before the 'n', correct?
Does this mean this could work with any numbers for 'a' and 'n', as long as they are below the previously mentioned values? Will it simply produce different angles? I don't have a protractor just yet to play around with this and make it visual which always helps me out.

a has to be less than 12.85 (not 14 as originally stated), because with 7 divisions, 90/7=12.85.

You can't use any number for 'a' and 'n'. You can use any number for one of them (within the limits) and calculate what the other is.

I actually have no idea what sup stands for, but am going to look it up!

I didn't realise I was this rusty with math haha, it's fun getting back into it though!

Sorry, on my screen [sup] doesn't show up. It is forum html for superscript. x-squared = x²

Fret spacing uses this same algorithm

[popinfresh]

Originally Posted by Frank Tuesday

Divide the numerator and denominator both (top part of fraction) by 3,

a has to be less than 12.85 (not 14 as originally stated), because with 7 divisions, 90/7=12.85.

You can't use any number for 'a' and 'n'. You can use any number for one of them (within the limits) and calculate what the other is.



Sorry, on my screen [sup] doesn't show up. It is forum html for superscript. x-squared = x²

Fret spacing uses this same algorithm

Sorry, I totally missed your reply! Thanks heaps, I can wrap my head around it now
One last question, what did you use to draw up the pictures with different angles?