Here is a guide for a genuine mathematical experiment that you can carry out at home. In order to do this, however, you will need an iPad. (It might also work with a suitable app on a different tablet, but I was not able to test this.)
As well as the iPad you will need a room with a mirror - a bathroom is ideal for this.
On the iPad there is an app called PhotoBooth which allows you to take selfies. In general this comes pre-installed on the device. In this app there is a photo mode called "kaleidoscope". Choose this option and switch on the selfie mode.
Now take the iPad into the room with the mirror. It works best when the room isn't too bright. You might also need to experiement with the brightness of the iPad screen, as the experiement tends to work better when the screen is a bit darker.
Hold the iPad with the screen facing the mirror so that the iPad can "see" itself. Keep adjusting the position until the iPad camera sees its own screen in the mirror. When the iPad can see its own screen, this produces an effect called an optical feedback loop. The image is reflected and is recorded in a distorted way, and this is then reflected again in another image. In this way crazy patterns are created.
Sometimes you might need a little flash of light (for example opening the door for a short time) to get it working. When patterns do appear, you can transform them by giving the iPad a little shake.
The following video was recorded one evening in a bathroom with low light and with the iPad screen's brightness turned down. Apart from adding the music, the video has not been edited at all.