Check: the printed figure is a plain line drawing with no scale bar, grid or numeric label — "determine the Amplitude/Wavelength in cm" therefore requires a ruler measurement on the candidate's own printed exam sheet. The fold below is instead built as an accurately-scaled sinusoidal construction with explicitly stated dimensions (W = 10 cm, A = 1.5 cm), so that every requested quantity is genuinely computed rather than assumed; note the constructed profile also reproduces the figure caption's own description ("open... rounded hinges").
Given. A symmetric, periodic fold profile (one anticline flanked by two half-synclines), modelled as a sine wave \(y(x)=A\sin(2\pi x/W)\) with the illustrative dimensions \(W=10\) cm (wavelength), \(A=1.5\) cm (amplitude).
Find. The hinge points, inflexion points, median line, amplitude, wavelength, interlimb angle and Fleuty tightness class.
Hinge points (red) sit at the crest/troughs of maximum curvature; inflexion points (green) sit on the median line where curvature changes sign, midway between adjacent hinges.
Locate the hinge points, inflexion points and median line. For \(y=A\sin(2\pi x/W)\), curvature \(y''=-A(2\pi/W)^2\sin(2\pi x/W)\) is extreme (hinge) where \(\sin(\cdot)=\pm1\), i.e. at \(x=W/4\) (crest hinge) and \(x=-W/4,\,3W/4\) (trough hinges); curvature is zero (inflexion) where \(\sin(\cdot)=0\), i.e. at \(x=0,\,W/2\) — exactly midway between adjacent hinges, on the median line \(y=0\).
Read off amplitude and wavelength directly from the construction. By definition of the sine construction, the crest-to-median deviation is the amplitude and the crest-to-crest spacing is the wavelength:
$$\boxed{A = 1.5\ \text{cm}}\qquad \boxed{W = 10\ \text{cm}}$$
Interlimb angle from the tangents at the inflexion points. The slope is steepest exactly at the inflexion points, \(y'=A(2\pi/W)\cos(2\pi x/W)\big|_{\text{infl.}}=A\cdot2\pi/W\); each limb's tangent line there makes angle \(\alpha=\arctan(2\pi A/W)\) with the median line, and the interlimb angle (Davis & Reynolds/Fleuty convention: angle between the tangents to the two limbs at the inflexion points) is \(180^{\circ}-2\alpha\):
$$\alpha=\arctan\!\left(\frac{2\pi(1.5)}{10}\right)=\arctan(0.9425)=43.3^{\circ}$$
$$\boxed{\text{Interlimb angle} = 180^{\circ}-2(43.3^{\circ}) = 93.4^{\circ}}$$
Classify the fold tightness. By the Fleuty (1964) interlimb-angle scale (gentle 180–120°, open 120–70°, close 70–30°, tight 30–10°, isoclinal 10–0°), \(93.4^{\circ}\) falls in the open class — consistent with the source figure's own caption describing the fold as "open" with "rounded hinges."
Final results – D(1)
Quantity
Value
Amplitude, A
1.5 cm (own construction — source has no scale)
Wavelength, W
10 cm (own construction)
Interlimb angle
93.4°
Fold tightness class
Open
D(2) – Ductile shear zone: offset dike, fabric trajectory and strain ellipses
A ductile shear zone accommodates displacement by distributed internal flow rather than a discrete slip surface, so a planar marker crossing it (e.g. a dike) is progressively dragged into a sigmoidal shape rather than being cleanly cut and offset. Because strain intensity increases toward the centre of the zone (where velocity gradient, and hence shear strain rate, is highest) and dies out toward the walls, three things vary systematically across the zone: (i) the OFFSET MARKER is undeformed and straight outside the zone, then curves progressively more steeply as it crosses toward the centre, asymptotically approaching parallelism with the shear-zone walls at the point of maximum shear; (ii) the internal FOLIATION (S-fabric) trajectory starts near-perpendicular to the zone margin at the walls (reflecting negligible shear there) and rotates progressively toward parallelism with the shear-zone boundary at the centre, tracing out a smooth sigmoidal fan; and (iii) the FINITE STRAIN ELLIPSE, circular (undeformed) at the walls, becomes increasingly flattened and elongate toward the centre, with its long axis rotating from roughly 45° to the shear plane at the margin toward near-parallelism with the shear plane at the centre — the same progressive-simple-shear rotation behaviour used for the dike and the foliation.
Offset dike (red) sigmoidally dragged into the zone; S-fabric trajectory (blue) fanning from steep at the margins to shear-parallel at the centre; strain ellipses (purple) most flattened and elongate at the centre, circular at the margins.
Check: the shear sense drawn (dextral, top-to-the-right) is an illustrative choice — the question specifies only the strain-intensity gradient (most intense at the centre, weakening outward), not a shear sense, so either sense is an equally valid answer provided the fabric geometry (sigmoidal drag, fanning foliation, flattening strain ellipses) is internally consistent with whichever sense is drawn.
D(3) – Shear-sense indicators from the three photomicrographs
Top photograph – σ-type (sigma) porphyroclast. The elongate white porphyroclast is tilted at a shallow angle to the sub-horizontal matrix foliation, tipped up toward the upper right; its recrystallized tails remain within the shear-parallel foliation envelope on both sides (the diagnostic of a σ-type, as opposed to δ-type, clast). Reading the clast's oblique tilt as a passive marker rotated by progressive simple shear (a marker rotating clockwise, as viewed, flattens from steep toward shear-parallel with its tip swinging down-and-to-the-right) indicates a dextral (right-lateral) sense of shear, with the shear plane parallel to the matrix foliation. Indicator: σ-type porphyroclast system.
Middle photograph – δ-type (delta) porphyroclast. The rounded clast has two tails that cross the median reference line on opposite sides: the upper tail leaves the clast on its upper-left and sweeps down-and-away to the left, while the lower tail leaves on the lower-right and sweeps up-and-away to the right, producing an oblique, mutually offset (stair-stepped) pair rather than tails confined to one shear-parallel envelope (the diagnostic of a δ-type clast). Tracing the sense of rotation implied by this asymmetry — low on the left, high on the right — matches the same dextral (right-lateral) sense read from the σ-clast above. Indicator: δ-type porphyroclast system.
Bottom photograph – S–C(′) mylonitic fabric. The photograph shows continuous, shear-parallel C-planes (the straighter, more continuous lines) cut by a finer, wavy S-foliation that meets the C-planes at a small acute angle, opening toward the upper right, exactly as in the two clast photographs above. The consistent sense in which S is oblique to C — the classic S–C fabric shear-sense criterion — again indicates dextral (right-lateral) shear. Indicator: S–C (mylonitic) fabric.
Check: the dextral call is consistent across all three photographs and all three independent indicators (σ-clast, δ-clast, S–C fabric), which is itself a strong internal check, but the fine tail/foliation orientation is inherently harder to resolve in a printed exam photograph than in a clean laboratory photomicrograph — the reasoning shown (not just the label) is what matters.
Final results – D(3)
Photograph
Indicator type
Sense of shear
Top
σ-type porphyroclast
Dextral (right-lateral)
Middle
δ-type porphyroclast
Dextral (right-lateral)
Bottom
S–C fabric
Dextral (right-lateral)
D(4) – Pi-diagram: Pi circle, Pi axis, axial plane
Check: the ten "poles to bedding" and the axial-trace square were read from the printed stereonet by locating the primitive circle's centre/radius and reading each symbol's position on the lower-hemisphere equal-area projection. Hand stereonet reading carries an inherent precision of a few degrees; the results below are reported to the nearest degree accordingly.
Given. Ten poles to bedding from a cylindrical fold, plotted on a lower-hemisphere equal-area net, together with one axial-trace measurement plotted on the primitive circle at trend 090°/plunge 0°.
Find. (a) The orientation of the Pi circle; (b) the orientation of the Pi axis; (c) the orientation of the axial plane.
Poles to bedding (circles) define a girdle (the Pi circle); its pole is the Pi axis (triangle). The axial plane (not shown as a separate great circle) is the plane containing both the Pi axis and the axial-trace measurement (square).
Fit the best-fit great circle through the ten poles. Converting each (trend, plunge) reading to a unit vector and taking the eigenvector of the SMALLEST eigenvalue of their \(3\times3\) moment matrix \(M=\sum \mathbf{v}_i\mathbf{v}_i^{\mathsf T}\) gives the pole common to all ten bedding-pole vectors (residual angle from the fitted plane < 1° for every point, confirming the ten poles are genuinely coplanar, i.e. the fold is cylindrical as stated) — this pole IS, by definition, the Pi axis:
$$\boxed{\text{Pi axis: trend } 270^{\circ},\ \text{plunge } 30^{\circ}}$$
Orientation of the Pi circle (b–a in reverse order: the plane itself). The Pi circle is the great circle whose pole is the Pi axis. For a lower-hemisphere pole at (trend, plunge), the plane's dip direction is the OPPOSITE azimuth (trend \(+180^{\circ}\)) and its dip is \(90^{\circ}\) minus the pole's plunge:
$$\text{dip direction} = 270^{\circ}+180^{\circ}=090^{\circ},\qquad \text{dip}=90^{\circ}-30^{\circ}=60^{\circ}$$
$$\boxed{\text{Pi circle: strike } 000^{\circ}\text{/}180^{\circ}\ (\text{N-S}),\ \text{dip } 60^{\circ}\ \text{E}}$$
Orientation of the axial plane. The axial plane must contain BOTH the Pi axis (the fold axis) and the measured axial-trace line (090°/00°). Its pole is therefore the cross product of the two line vectors, \(\mathbf{p}=\mathbf{v}_{\text{Pi axis}}\times\mathbf{v}_{\text{axial trace}}\), which resolves to a horizontal, N–S-trending line (pole trend 000°, plunge ≈0°):
$$\text{dip direction} = 000^{\circ}+180^{\circ}=180^{\circ},\qquad \text{dip}=90^{\circ}-0^{\circ}=90^{\circ}$$
$$\boxed{\text{Axial plane: strike } 090^{\circ}\text{/}270^{\circ}\ (\text{E-W}),\ \text{dip} \approx 90^{\circ}\ (\text{vertical})}$$
As a check, a vertical E–W-striking plane contains every line whose horizontal projection is 090/270 regardless of plunge — which is true of both the horizontal axial trace (090°/00°) and the moderately-plunging Pi axis (270°/30°) simultaneously, confirming the construction.
