I have posted online some
matlab scripts, GMT scripts, and mathematica scripts
relevant for interpreting and plotting moment tensors. These
files are available within the compearth
repository on github. They can be downloaded from github as a zipped file or using git from the command line. Please email me with corrections or suggestions. 
(a) The fundamental lune L [orange], the
same lune as in Fig. 1. Each point on the lune is a
beachball pattern (moment tensor source type). The
isotropic axis is vertical, hence the deviatoric plane
(purple) is horizontal. The three green planes are the
mirror planes—the planes of symmetry for the three
permutations that are transpositions. Each of the six
lunes delineated by the mirror planes corresponds to an
ordering of λ1, λ2, λ3, and each could serve as a
fundamental lune. The upper blue arc is λ3 = 0; on and
above it all beachballs are red. The lower blue arc,
nearly out of sight, is λ1 = 0; on and below it all balls
are white. Seven beachballs are shown, all on the same
meridian γ = −10. Balls from top to bottom have latitude δ
= 90, 47, 20, 0, −20, −53, −90. (b) Five deviatoric beachballs. The plane
of the screen is the deviatoric plane. [Tape and Tape,
2012a, Fig. 11] 
Home for beachball patterns (moment tensor source types)—the fundamental lune of the unit sphere. Each point on the lune represents a beachball pattern. The magenta equatorial arc is for deviatoric tensors, the red meridian is for sums of double couple and isotropic tensors, and the black arc is for crack + double couple tensors having Poisson ratio ν = 1/4. Above the upper blue arc all beachballs are only red, and below the lower blue arc all balls are only white. ISO, isotropic; DC, double couple; LVD, linear vector dipole; CLVD, compensated linear vector dipole; C, tensile crack with Poisson ratio ν = 1/4. [Tape and Tape, 2012a] 
Five published compilations
of full moment tensors, represented on the fundamental
lune. The dashed line is the crack + double couple arc for
ν = 1/4. [Tape and Tape, 2012a] 
Beachballs plotted on the 2D version of the
fundamental lune. The double couple is at the center; its orientation changes for each example. 
Four regimes
for moment tensors. Above and to the right of the arc
lam2=0 (blue), moment tensor beachballs have red bands and
white caps (lam2 > 0 > lam3). Below and two the left
of it they have white bands and red caps (lam2 > 0 >
lam2). Above the arc lam3=0 (red), beachballs are all red
(lam3 > 0). Below the arc lam1=0 (white), they are all
white (lam1 < 0). [Tape and Tape, 2013, Fig. S1] 
(left) Some
moment tensors with middle eigenvalue = 0. The angle
alpha (plotted as orange contours) is the angle between
the fault normal vector and the slip vector. For moment
tensors with lambda2 = 0, alpha gives the angular extent
of either of the white lunes of the beachball. (right)
Some crack tensors, which lie on the boundary of the
lune. [Tape and Tape, 2013] 
phi = 15 (zeta = 90) 
phi = 45 (zeta = 90)  phi = 75 (zeta = 90)  phi = 105 (zeta = 90) 
Alternative depiction of four of the
beachballs represented in Tape and Tape (2013), Figure
6. (These figures were plotted using a Mathematica notebook available in the compearth github repository.) 
Any moment
tensor (M) can be expressed a linear combination of two
orthogonal tensors: a crack tensor (K) and a double
couple (D) such that the crack plane of K is a shear
plane of D (Minson et al., 2007). A key point is that K
and D do not have the same basis. The angle zeta
measures the amount of K (crack) within M; it ranges
from zeta = 0 at the center (D) to zeta = 90 at the lune
boundary. note: zeta is not the same as the arc distance
to the lune center point. [Tape and Tape, 2013] 
Previously
published moment tensors of real events, here plotted on
the lune. [Tape and Tape, 2013] 
Previously
published moment tensors for induced events, here
plotted on the lune. The line between (1,0,0) and
(0,0,1) marks the subset of moment tensors with middle
eigenvalue = 0. [Tape and Tape, 2013, Fig. S14] 
The
uniform parameterization of moment tensors. The
parameterization handles moment tensor eigenvalues
(top row) separately from orientation parameters
(bottom row). [Tape and Tape, 2015, Fig. 1] 
Lune with
contour plot of probability density for eigenvalue
triples $(\backslash lambda\_1,\backslash ,\backslash lambda\_2,\backslash ,\backslash lambda\_3)$
of moment tensors. The density is maximum for double
couples (center of lune) and falls off to zero at the
lune boundary, where $\backslash lambda\_2\; =\; \backslash lambda\_1$
or
$\backslash lambda\_2\; =\; \backslash lambda\_3$.
[Tape and Tape, 2015, Fig. 2] 
