rotations of cubes give examples of group operations and also of homomorphisms if you look at the action on axes, or other subsets.
in fact it is my oopinino tha essemtiwlly abstratc concepts arose from concrete examples, so it should be possible to find them for anything.
abstract divisibility of integers arose in Greek times as the theory of measurement. I.e. they said one unit "measures" another instead of divides it, which clearly means that a meter stick can be used to emasure precisely those lnegths whose length is divisible by one meter.
indeed the familiar proof that all ideals of integers are priniple folows fro this concrete analogy a follows:
the usual abstartc proof says to atke an element of the dieal of shortest length nd them use the divison algorithm and contradiction to show it divides all other elements.
concretely one looks at any finite collection of elements on the line and then consider two other lengths, namely the shortest length that can be measured using those given oens as measuring sticks, and second th longest length that can be used alone as a measure to measure all the given ones, assuming these exist. they exist of course if and only if the given lengths are "commensurable" which the greeks initially assumed always occurred.now we claim that given two commensurable lnegths on the line, that the two lengths are the same, i.e. the longest length that will measure both of them, rquals the shortest length they can measure together.
to see this consider all points of the line they can measure together (i.e. all integer linear combinations of the two given integer multipels of some "unit" length).
so you are given two measuring sticks and using them and some given starting point, draw in all points whose location can be exactly measured from that starting point using those two sticks.
assume there is a shortest one. now we claim that all those points are equidistant, i.e. they are all multiples of that on shortest one.
to see this note that if two points can be measured so can their difference and sum. hence if the distnace between A and B were shiorter than tht betwen P and Q, we could measure off the shorter distnace betwen A and B and then ad it to P to get somewhere between P and Q.
Now that all succesive pairs of points are equidistant, they must all be multiples of the segment from the starting point to the first new point. hence that shortest distance that both sticks can measure, is also the longest length that will measure all the others, including the two originally given ones.
so the abstract proof that every ideal of integers is principal is equivalkent to the geomnetric fact that the shortest distnace measurable by two measuring sticks, equals the longest distnace that will measure both of them.
the conceopt of a "neighborhood" I am topology is the same as the commonly used one in everyday speech. once a builder friend of mi e said he had made someone an offer, "between 475 and 500, somewhere in that nighborhood" then he remarked to me it was not a very large neighborhood, so i told him that as exactly what the term means in math, that one srudies things precisely, by repeatedly approximating them, and in fact one specifies smaller and smaller neighborhoods of the desired value.
Here's an concrete analogy for expressing the idea of an Archimedean field.
