1. Can we quantitatively relate LPO fabrics to deformation symmetry? Relating Lattice Preferred Orientation to Deformational Process using Statistical Analysis of Symmetry in Orientation Distribution Space Christopher Thissen Mark Brandon Yale University

2. Symmetry in Quartz LPOs Thigpen et al., 2010 c- axes a-axes n=101114 Can we improve on the skeleton diagram approach by making quantitative estimates about the fabric symmetry?

3. Curie’s Symmetry Principle – The symmetry of the effect is at least as great as the symmetry of the cause. – So: The symmetry of the LPO is at least as high as the symmetry of the factors causing LPO and is usually directly related Quantitative Links between LPO and Deformation Symmetry Curie, 1894 L Foliation (S)

4. Uniaxial Orthorhombic Monoclinic Girdle of rotation axes Quantitative Links between LPO and Deformation Symmetry Quartz c-axes

5. ABOVE: Consider a monomineralic rock with an initial random orientation distribution of a hypothetical cubic mineral, with a single slip system, with slip in the (001) plane in the [100] direction. The initial stereograms for [100], [010], and [001] would all look the same, like the example above. LEFT: The three stereograms show the orientation distribution for the three axis directions [100], [010], and [001] after a coaxial deformation with strain directions X, Y, and Z. Z X Z X Initial Distribution X X [100] [010] [001] Final Orientation Distribution 5

6. [100][010] [001] Z X Quantifying Symmetry Searching for Symmetry Operators X X X Black: Original Distribution Blue: Rotated Distribution (rotation at 000,00) Good FitPoor Fit

7. [100][010] [001] Z X Quantifying Symmetry Searching for Symmetry Operators X X X Black: Original Distribution Blue: Rotated Distribution (rotation at N) Good FitPoor Fit

8. [100][010] [001] Z X Quantifying Symmetry Searching for Symmetry Operators Z X X Z X Z Black: Original Distribution Blue: Rotated Distribution (rotation at 045, 00) Good FitPoor Fit

9. [100][010] [001] Z X Black: Original Distribution Blue: Rotated Distribution (rotation at 045, 00) Quantifying Symmetry Searching for Symmetry Operators Z X X Z X Z Good FitPoor Fit

10. [100][010] [001] Z X Quantifying Symmetry Searching for Symmetry Operators Z X X Z X Z Good FitPoor Fit

11. Quantifying Symmetry Searching for Symmetry Operators The orientation of each crystal can be characterized by three rotation angles, called Euler Angles. The Euler angles can be used to define a 3D space in which each crystal orientation is represented by a point. Engler and Randle, 2010

12. Quantifying Symmetry Searching for Symmetry Operators Reduced Chi^2 Value 0 50

13. Quantifying Symmetry Synthetic Olivine Example N=500 [100][010] [001] Reduced Chi^2 Value 0 50

14. Quantifying Symmetry Synthetic Olivine Example N=500 Rotation Axes [100][010] [001] 0 50 Reduced Chi^2 Value

15. Quantifying Symmetry Synthetic Olivine Example N=500 Rotation Axes [100][010] [001] 0 50 Reduced Chi^2 Value

16. Quantifying Symmetry Moine Thrust Mylonites Law, 2010

17. Quantifying Symmetry Moine Thrust Mylonites Law, 2010

18. Quantifying Symmetry Moine Thrust Mylonites Thigpen, et al., 2010 c- axes a-axes MT-07-21 n=10,000 4 14

19. Conclusions We can statistically quantify the symmetry of LPO fabrics using Chi^2 in Euler space The goal is to quantify the type of symmetry present and the symmetry orientation and relate this to the deformation