It's not a "Sun Phenomenon".
The hole is acting as a pin hole camera. So as the man says in the video the image on the wall is an image of the sun NOT an image of the hole. If the sun were square the image on the wall would be square regardless of the shape of the hole.
You know about pinhole cameras, right? So with a really tiny hole, what shape would you expect to see? What would determine the size of it? Now think of a slightly larger hole as many small holes. Each will make an image like the single small hole case, but they won't overlap perfectly. There will be a central bright region, where they do overlap, and a progressively darker annulus. What will determine the size of the annulus? Under what circumstances will the new image, including annulus, be only slightly larger than the original?
After doing more research I understood how it worked. I understand that the light comes from many different angles and then forms an upside down image. However, in that video, what if we ignore the fact that light from the sun can bounce off objects? Then, the image would just be a square instead of a circle, right?
It's nothing to do with the light bouncing.
The sun's disc subtends an angle of about half a degree at the Earth, or 0.01 radians. If a card with a pinhole is 0.5m from its shadow, the sun's image will be about 5mm across. If you make the hole bigger, the effect is to superimpose multiple sun images. The centres of the images will map out exactly the same shape and size as the hole. So if the hole is less than 3mm across you will still see an almost circular image, and only a little larger.
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