Ants use a combination of vector addition and root mean square to find their way home.
When they move they estimate the direction (from the sun) and their distance travelled (how many steps). They add these vectors to always have a "home that way" vector. Unfortunately small errors in either the distance or direction could lead to the ant not being home when she thinks she is. What happens then?
An ant learns the appearance of the environment around the home. Her eyesight is low resolution so she can not easily recognise home in the same way as we can by high resolution appearance, plus her environment is mostly trees and more trees.What she does is processes each new image in comparison to these images she has of home. She does the equivalent of a root mean square of the difference between the pixels in the new image compared with one of her stock images. She moves, and recalculates. If the value is tending towards zero, she continues in that direction. If not, she tries a different route. When the value is zero, she is home.
http://plus.maths.org/content/finding-way-home
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