Vizualizing Infinity: Exploring Non-Flat Surfaces

In summary, the conversation discusses a process of drawing circles within circles on a plane and repeating it a million times, with the number of points and the radius tending to infinity and a very small non-zero quantity, respectively. It is mentioned that for certain values of n, the resulting image looks like a graph, while for others it fills the entire plane. The possibility of creating a fractal by halving the radius is also mentioned, but it is noted that the algorithm currently grows exponentially.
  • #1
exmachina
44
0
Try drawing this mentally:

Start with a circle of radius r, draw n number of points spaced evenly on the circle. at each point on the circle draw another circle of radius r, once again with n number of points. What sort of a picture would one get repeating this process a million times, and as n tends to infinity and r tends to some very small non-zero quantity?

I was told that: for n = 1, 2, 3, 4, or 6, it looks like a bunch of lines connected together at points in a regular way, like an infinite graph. For any other n, it just fills the whole plane.

But I still can't visualize it. This is a preliminary to an extension of non-flat surfaces (ie. repeat the process only this time on a riemannian manifold)
 
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  • #2
exmachina said:
Try drawing this mentally:

Start with a circle of radius r, draw n number of points spaced evenly on the circle. at each point on the circle draw another circle of radius r, once again with n number of points. What sort of a picture would one get repeating this process a million times, and as n tends to infinity and r tends to some very small non-zero quantity?

I was told that: for n = 1, 2, 3, 4, or 6, it looks like a bunch of lines connected together at points in a regular way, like an infinite graph. For any other n, it just fills the whole plane.

But I still can't visualize it

Why not take a compass and paper and try it?
 
  • #3
I did, but it becomes very hard to do as n gets large, especially in the limit as n tends to infinity
 
  • #4
If you half the radius of the circle during each iteration, you'd get some kind of fractal on the boundary.
 
  • #5
the radius r is constant
 
  • #6
I don't see a pattern in anything other than n=2,4. Put the center of the circle at the origin. If you take the example n=5, the x and y coordinates of four of the points contain pi. I don't see anything other than noise comming out.
 
  • #7
So I wrote a program , it definitely does seem to pack for n = 1,2,3,4,6

#include <stdio.h>
#include <SDL/SDL.h>
//#include <SDL/SDL_draw.h>
#include <math.h>

#define WIDTH 4000
#define HEIGHT 2000
#define BPP 4
#define DEPTH 32
#define ORIGINX WIDTH/2
#define ORIGINY HEIGHT/2

#define NUMPOINTS 5
#define RADIUS 45
#define PI 3.1415926
#define SEARCHDEPTH 11

void setpixel(SDL_Surface *screen, int x, int y, Uint8 r, Uint8 g, Uint8 b)
{
Uint32 *pixmem32;
Uint32 colour;

colour = SDL_MapRGB( screen->format, r, g, b );
int ytimesw = y*screen->pitch/BPP;


pixmem32 = (Uint32*) screen->pixels + ytimesw + x;
*pixmem32 = colour;
}

void drawcircle(SDL_Surface *screen, int centerx, int centery, int iter)
{

int deltax;
int deltay;

double dangle = 2*PI/NUMPOINTS; // starts at pi/3
double angle;

for(angle = 0; angle < 2*PI ; angle+=dangle)
{

if (iter>=SEARCHDEPTH) {
return 0;
} deltax = (int) RADIUS*cos(angle);
deltay = (int) RADIUS*sin(angle);

setpixel(screen, centerx+deltax, centery+deltay, 255, 255, 255);
// Draw_Circle(screen, centerx+deltax, centery+deltay, RADIUS, 255);
/*DEBUG
printf ("cos of angle %f\n", cos(angle));
printf ("angle: %f\n", angle);
printf ("deltax: %d\n", deltax);
printf ("deltay: %d\n", deltay);
printf ("iteration: %d\n", iter);
*/

//updates the screen only 1/5th of the time
if (iter==SEARCHDEPTH/2){
SDL_Flip(screen);
}

iter++;
drawcircle(screen, centerx+deltax, centery+deltay, iter);
iter--;


}

}

void DrawScreen(SDL_Surface* screen, int h)
{

if(SDL_MUSTLOCK(screen))
{
if(SDL_LockSurface(screen) < 0) return;
}

drawcircle(screen, ORIGINX, ORIGINY, 0);

if(SDL_MUSTLOCK(screen)) SDL_UnlockSurface(screen);

}int main(int argc, char* argv[])
{
SDL_Surface *screen;
SDL_Event event;

int keypress = 0;
int h=0;

if( SDL_Init( SDL_INIT_EVERYTHING ) == -1)
{
return 1;
}

screen = SDL_SetVideoMode(WIDTH, HEIGHT, DEPTH, SDL_SWSURFACE);

if (screen == NULL )
{
return 1;
}

SDL_WM_SetCaption( "Drawing the 'circles' ... ", NULL );

//DrawScreen(screen,h++);

while(!keypress)
{
DrawScreen(screen,h++);

while(SDL_PollEvent(&event))
{
switch (event.type)
{
case SDL_QUIT:
keypress = 1;
break;
case SDL_KEYDOWN:
keypress = 1;
break;
}
}
}

SDL_Quit();

return 0;
}
 
  • #8
wanna post some screenshots? I am curious
 
  • #9
here's what it looks like for 3 iterations, the intensity scale is from dark red (very few points) to yellow (medium # of points) to white ( a lot of points). note that right now, the algorithm is quite terrible. Unfortunately it grows exponentially (ie. n^i n = number of points, i = iterations) so 50 points per circle, 5 iterations, 312 million points. I haven't found any smart ways to break the exponentiality yet.
 

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FAQ: Vizualizing Infinity: Exploring Non-Flat Surfaces

1. What is "Vizualizing Infinity: Exploring Non-Flat Surfaces"?

"Vizualizing Infinity: Exploring Non-Flat Surfaces" is a scientific concept that explores the idea of visualizing and understanding infinite shapes and surfaces that are not flat or two-dimensional. It is a field of study that combines mathematics and art to create a better understanding of the concept of infinity.

2. Why is it important to study non-flat surfaces and infinity?

Studying non-flat surfaces and infinity is important because it allows us to expand our understanding of the world around us. Many natural phenomena, such as the shape of the universe and the growth patterns of plants, cannot be fully understood without considering the concept of non-flat surfaces and infinity. Additionally, studying these concepts can have practical applications in fields such as architecture and engineering.

3. How do scientists visualize infinity and non-flat surfaces?

Scientists use a variety of methods to visualize infinity and non-flat surfaces. These can include mathematical equations, computer simulations, and physical models. Some scientists also use visualization tools such as virtual reality to better understand and represent these complex concepts.

4. What are some real-world examples of non-flat surfaces?

Some examples of non-flat surfaces in the real world include the surface of a sphere, the shape of a potato, and the curves of a rollercoaster. These surfaces may appear flat to the naked eye, but they have a curvature that can only be fully understood when considering the concept of infinity.

5. How does the study of non-flat surfaces and infinity impact other fields of science?

The study of non-flat surfaces and infinity has a significant impact on various fields of science, including physics, biology, and astronomy. It allows scientists to better understand the complex nature of the universe and the patterns and structures found within it. It also has practical applications in fields such as architecture and design, where understanding non-flat surfaces is crucial for creating stable and functional structures.

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