ways to define harmonic functions

There are at least three ways to define harmonic functions such as
sine, cosine, and the complex exponential:
1. as relations between sides and angles of triangles in plane
geometry

2. as solutions of second order homogeneous differential equations
with certain initial conditions as in
D^2 cos + cos = 0 with init.cond. cos(0)=1 [Dcos](0)=0
D^2 sin + sin = 0 with init.cond. sin(0)=0 [Dsin](0)=1
D^2 expi + expi = 0 with init.cond. expi(0)=1 [Dexpi](0)=i

3. as polynomials in a Taylor expansion

Does anyone know of any other definitions of harmonic functions on any
other mathematical structures?

Many thanks in advance,
Dan Greenhoe

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