## Animation of refold structures

Animation of refold structures

The animations show finite refold geometries of two folding phases having the same wavelength/amplitude ratio of 2. Two kinds of movies are presented: three-dimensional surface plots (referred to as structure-movies), and fold interference patterns on three perpendicular faces of a block oriented perpendicular to the kinematic axes of the initial fold (referred to as pattern-movies). The animations show the geometrical transition from one refold end-member into another - in 18 steps, with 5° difference each. Note that the orientation of the block models does not change and is identical through all animations. Therefore changes in the shape of the interference patterns result from different superposition geometries and not from changing section orientations. The layers shown in the structure-movies consist of a central layer (dark yellow), visualizing the refold structures, and blue and/or red orthogonal marker layers that were introduced as planes after the first folding event. These marker layers visualize the orientation of the second fold and are therefore crucial for recognizing and distinguishing (!) Type 0_{1}, 0_{2} and 0_{3} refolds. In the pattern-movies refolded marker layers creating interference patterns are shown in dark yellow and pink. Orthogonal marker layers are shown in blue.

Both the structure- and the pattern-movies additionally show an animated stereographic projection (referred to as refold-stereoplot) of the incremental orientation of the fold axes and axial planes of the initial and superposing fold (Fig. 3). In this refold-stereoplot the spatial orientation of the initial fold is always fixed: *f*_{1} is oriented horizontally W-E and the axial plane is vertical striking W-E with a pole *c*_{1} oriented horizontally N-S. Because the refold Types 1-3 can be transformed in their Types 0_{1}-0_{3} counterparts by simply rotating their axial plane around the superposing fold axis *b*_{2} this transformation between the end-member positions can be elegantly displayed by traces of *c*_{2} during rotation along either the periphery or along the N-S and W-E diameter of the refold-stereoplot (strictly speaking traces of *c*_{2} during rotation plot along small circles around b_{2} for any spatial orientation of the superposing fold):

(i) Type 1 refolds plot in the periphery (fold axis N and S, *c*_{2} E and W) and are transformed to Type 0_{1} refolds by translating *c*_{2} along the W-E diameter towards the centre of the refold-stereoplot. (ii) Type 2 fold axes plot in the centre of the diagram, *c*_{2} in the E and W. The refolds are transformed to Type 0_{2} refolds by moving *c*_{2} along the periphery of the refold-stereoplot in a N-S orientation. (iii) Type 3 fold axes plot in the E and W, *c*_{2} plots in the centre of the diagram. The refolds are transformed into Type 0_{3} refolds by translating *c*_{2} along the N-S diameter towards the periphery of the refold-stereoplot.

Note that although always two different end-member refolds are plotting at the same N-S, W-E or centre node of the refold-stereoplot, the structures are clearly distinguished by the orientation of their fold axis *b*_{2}. The following section gives a short description of the structure- and pattern-movies of all 15 possible combinations of the 6 end-member structures. Although such progressive transitions between the end-member types are purely geometrical we think that a careful study of these animations of developing refold shapes together with the wide range of possible interference patterns is a perfect training for the understanding of complex three-dimensional shapes and intersections occurring in nature. The movies are described in following logical groups:

The geometrical difference between the end-members requires rotating *c*_{2} around b_{2} by an angle of 90Á. Consequently the superposing fold axis does not change its orientation. A natural example of such transitions could be expected in polyphase deformed areas with a great variability in the orientations of the axial plane and a uniform distribution of the fold axis of the superposing folds (e.g. Fusseis, 2001). Note that, despite Type 0_{1} - 0_{3} do not produce any visible interference patterns, irregular interference patterns can be observed along most of the transition paths. Structures between Type 1 and 0_{1} will be characterized by dome-basin and/or pronounced banded s-z-shaped interference patterns, almost resembling an asymmetric crenulation cleavage with microlithons and cleavage domains (e.g. Passchier and Trouw, 1996). These banded s-z-shaped structures are typical and frequently found on two-dimensional sections in polyphase folded areas (Ramsay and Lisle, 2000). Such banded structures also occur on sections through structures between Type 2 and 0_{2}, where additionally crescent and w/m-shaped intersections are found. Interferences on sections normal to the fold axes from structures between Type 3 and 0_{3} are dominated by all variations of hooks and irregular convergent-divergent patterns. Note that this progressive transition between Type 3 and 0_{3} is the only model, where sections parallel to the fold axes would always record plane strain deformation.

These animations show the transitions between the classical end-members of fold interferences (Ramsay, 1967). Shapes like convergent-divergent hooks, dome and basins and dome-crescent-mushrooms patterns can be observed in the movies. However, note the great complexity and variety of the patterns even on sections orthogonal to the kinematic axes of the perfect cylindrical initial folds. Type 1 is transformed in Type 2 by rotation parallel to *c*_{2} and therefore the superposing axial plane does not change its orientation. Similarly the transition from the Type 2 into Type 3 requires a simple rotation around the orientation of *c*_{1}.

Whereas all previous examples could be transformed by a single rotation about one of the kinematic axes of the superposing fold, the transition of Type 1 into Type 3 needs either two kinematic rotation axes, or a single, oblique rotation axis that has to be constructed from the refold-stereoplot: (i) Find the great circle containing *b*_{2} of both Type 1 and Type 3 refolds and determine its pole Pb_{2}. (ii) Find the bisector of the angle between *b*_{2} of Type 1 and *b*_{2} of Type 3 refold. (iii) Draw a great circle between the bisector and Pb_{2} (iv) Repeat this construction for *c*_{2} of both Type 1 and Type 3 refolds. (v) The intersections between the great circles containing the bisectors represents the single oblique rotation axis transforming Type 1 into Type 3 refold structure.

This set of animations show the transition between Type 0 refold structures and emphasizes the importance of distinguishing between the suggested classes Type 0_{1}, 0_{2} and 0_{3}.

The transformations from Type 0_{1} into 0_{2} and Type 0_{2} into 0_{3} are again controlled by a simple rotation around one of the kinematic axes of the superposing fold (*a*_{2} and *c*_{2} respectively). On sections perpendicular to *f*_{1} no interference patterns are observed, and the sections normal to d_{1} and *c*_{2} show simple linear intersections throughout the transformation. However, folding of a marker layer intruded normal to *f*_{1} and/or to *d*_{1} clearly demonstrates the superposed heterogeneous deformation. If this superposition is not recognized, the assumption of plane strain in a cross section perpendicular to *f*_{1} is wrong and could lead to potential misinterpretations (e.g. when reconstructing balanced cross sections).

The transformation from Type into 0_{1} into 0_{3} is more complex and was performed using two rotations around *a*_{2} and *c*_{2}. A single oblique rotation axes can be constructed from the refold-stereoplot with the same method outlined above. Although both end-members show no interference patterns on orthogonal sections to the kinematic axes of the initial fold, the transition refolds create a broad spectrum of complex interference shapes, e.g. dome-basins, crescent, s/z, complex hooks and banded structures.

The remaining six transformations between end-members describe transitions from Type 1, 2 and 3 refolds to Type 0 classes excluding the simple rotations around *b*_{2} already described. The Type 1 into Type 0_{3} transformation results from a single rotation around the *a*_{2} axis. On a section perpendicular to this rotation axis regular dome-basin interference patterns (egg-carton structures) are progressively converted in en-echelon basin and domes (OÍDriscoll, 1962) until all refolds are cylindrical with parallel fold axes resulting in linear intersection on planes perpendicular to *d*_{1} and *c*_{1} respectively. The Type 3 into Type 0_{1} transformation is again based on a single rotation but contrary to the previous example around the *c*_{2} axis. On sections perpendicular to the *f*_{1} axis hooks are progressively "unfolded" and similar to the previous model result in cylindrical refolds with parallel fold axes and consequently in linear intersections on planes perpendicular to *d*_{1} and *c*_{1}.

The following transformations are more complex and require rotations of the superposing folds around oblique axis or again, as they were modelled for the movies shown, around two, orthogonal axes:

The transition from Type 1 into Type 0_{2} shows on the section normal to *d*_{1} changeovers from dome-basins into asymmetric mushroom shapes and banded s/z structures. Interference patterns on sections normal to *c*_{1} reveal an interesting succession from unfolding, asymmetric folding and again unfolding to linear intersection. The symmetric fold intersections on the section normal to *f*_{1} transform into dome-basins, which get progressively overprinted with hooks showing again symmetric folds after complete transformation into Type 0_{2} refolds.

The interference patterns on three sections perpendicular to the kinematic axes of the initial fold between the end-members Type 2 and Type 0_{1} are characterized by symmetric crescent mushrooms shapes normal to *d*_{1}, hooks normal to *f*_{1} and banded s/z and w/m shapes normal to *c*_{1}.

Whereas the interference patterns on two orthogonal sections of the transformation model between Type 2 and Type 0_{3} are very similar to patterns discussed in the previous two models, showing asymmetric mushroom shapes and banded s/z structures, the section perpendicular to *f*_{1} is striking complicated: banded structures reveal multifaceted changes in irregular hook shaped folds.

The transformation of Type 3 into Type 0_{2} creates rather similar interference patterns than Type 3 into Type 0_{1} or Type 0_{3}: Hook-shapes of the convergent divergent patterns on sections normal to *f*_{1} are progressively "unfolded" resulting in a sinusoidal intersections in the Type 0 end-members, but on other orthogonal sections only straight intersections can be observed. However, careful inspections of the movies reveal the distinct differences emphasizing the need to distinguish between Type 0_{1}, 0_{2} and 0_{3}. Even more important is the fact that only the transition into the Type 0_{3} end-member is plane strain and all other sections normal to *f*_{1} are markedly non-plane strain.

Given the striking complexity of interference patterns and their continuous transitions between the end-member refold structures, this short description of the movies is far from being complete. It is left to the reader to explore the great variability of interference patterns and to compare the results of the animations with shapes suggested by Thiessen (1986). It is very instructive to observe the development of the blue and red marker planes introduced to the models after the initial folding especially when Type 0 end-members are modelled. Without this marker planes it is impossible to distinguish between Type 0_{1}, 0_{2} and 0_{3}.