Similar to my previous post, this is my visual proof which suggests that 1/4 + 1/16 + 1/64… = 1/3. Each term in the series is a quarter of the previous one. Each red area is a quarter the size of the one before it, and the final red area is converging to a third of the big triangle.
![matthen:
Here is a visual proof that 1/3 + 1/9 + 1/27 + … = 1/2 (each term is a third of the one before it.) This is shown by finding red areas, each one of which is a third of the area of the one before it. To split a triangle into thirds, you can simply find the midpoint and cut from there out to the three corners. At the end of the animation, we see that we are filling in the left half of the big triangle with red, which suggests the infinite sum is equal to a half. [code] [more]](http://24.media.tumblr.com/tumblr_llb7bj99HO1qfg7o3o1_400.gif)
Here is a visual proof that 1/3 + 1/9 + 1/27 + … = 1/2 (each term is a third of the one before it.) This is shown by finding red areas, each one of which is a third of the area of the one before it. To split a triangle into thirds, you can simply find the midpoint and cut from there out to the three corners. At the end of the animation, we see that we are filling in the left half of the big triangle with red, which suggests the infinite sum is equal to a half. [code] [more]
created with Agony
created with Agony
![matthen:
Another cross-eye 3D image (cross your eyes so you see on 3D image in the middle; help). This shows the twisting of a flat band of paper into a Möbius strip- an interesting surface as it has only one side and one edge; it is ‘non-orientable’. [more] [my code]](http://25.media.tumblr.com/tumblr_ll37dcRVIL1qfg7o3o1_400.gif)
there does not exist…
![matthen:
The function y=|x| is continuous, but not differentiable at x=0, because it makes a 90 degree turn there. Would it be possible to construct a function which is continuous, but differentiable nowhere? Bolzano’s function is an example of such a curve. The animation shows how it is created by repeating a certain step, then zooms in to show it is infinitely detailed. Jaggy on all scales. [more]](http://25.media.tumblr.com/tumblr_lkvub1olqU1qfg7o3o1_400.gif)
The function y=|x| is continuous, but not differentiable at x=0, because it makes a 90 degree turn there. Would it be possible to construct a function which is continuous, but differentiable nowhere? Bolzano’s function is an example of such a curve. The animation shows how it is created by repeating a certain step, then zooms in to show it is infinitely detailed. Jaggy on all scales. [more]
This is a stereoscopic image
This is a stereoscopic image
This is a stereoscopic image
This is a stereoscopic image
bubble chamber by complexification
(click on the pict to start the javascript (processing) applet…drawing in real time)
