3 layer orientation (Fig. 1d) and equation (2). 26 Again, we can determine the length of the largest interference pattern from GLYPH<1> k using the moir´e reciprocal lattice vectors from the FT, Thus, we calculate a value of D$_{3}$ = 24 : 7 GLYPH<6> 1 : 6 nm for the size of the largest observed pattern. 26 The uncertainty is due to a distortion in the hexagonal shape of the pattern, and is likely caused by strain, which will be discussed later. As shown in the schematic in Fig. 1d, the top layer is nearly aligned with the third. This creates two patterns of nearly equal size and orientation. The angle between the two moir´e patterns is small, producing a large interference pattern. In the case of Fig. 1e, we have applied a similar analysis to determine the orientation of the top three lattices. The small- est k vectors in Fig. 1f were determined to correspond to the moir´e pattern of the first and second layers by comparing the moir´e pattern orientation with respect to the reciprocal lattice vectors. We propose that there is a small rotation between the second and third layer. The interference of these two moir´e patterns is responsible for the hexagonal set of spots on the dotted circle in the FT (Fig. 1g). Consequently, the image in Fig. 1e looks qualitatively very di GLYPH<11> erent than the one in Fig. 1a. As further proof of concept, we image the pattern at mul- tiple sample biases for a constant tunneling current. Because the imaging is done in a constant current mode, the tip will move closer to the surface as the sample bias ( V$_{S}$ ) is de- creased. This technique is used in imaging Bernal stacked graphene bilayers, and shows a transition in the imaging of the atomic lattice. 27 At higher biases, every atom is imaged; however, as the bias is decreased the tip instead images every other atom. Figures 2(a-d) show the result of imaging at multiple bi- ases on a region within the image shown in Fig. 1a. At V$_{S}$ = GLYPH<0> 0 : 5 V, a single set of moir´e maxima is seen, corresponding to a moir´e pattern from the rotation of the first / second layer. As the bias decreases, the tip begins to image a second moir´e pat- tern. By V$_{S}$ = GLYPH<0> 0 : 05 V, the tip images only the second moir´e pattern, which apparently results from rotation of layers 2 and 3. From measured deviations in the expected exponential de- cay of tunneling current vs tip-height ( I vs z ), we believe that by V$_{S}$ = GLYPH<0> 0 : 05 V the tip is in contact with the surface. The dif- ference in tip height between V$_{S}$ = GLYPH<0> 0 : 5 V and V$_{S}$ = GLYPH<0> 0 : 05 V is GLYPH<1> z = 0 : 39 nm. The sample biases at V$_{S}$ = GLYPH<0> 0 : 3 V ; GLYPH<0> 0 : 2 V (Fig. 2b,c) show the transition in imaging the top moir´e to the one underneath. The tip bias, tunneling current and tip shape are known to have a large e GLYPH<11> ect on the imaging of a moir´e pattern. 10 This behavior was calculated for a rotated layer on top of AB stacked graphite 28 and e GLYPH<11> ects similar to our own observations were predicted; however, at present we do not understand how the moir´e patterns of this multilayer system depend on tip-sample distance. Relative strain induced between two stacked hexagonal lat- tices can have a large e GLYPH<11> ect on observed superlattice wave- lengths. This phenomenon is well known in moir´e interfer- ometry as a technique for strain measurement, 3 and has been observed in large period moir´e patterns on HOPG. 6 The e GLYPH<11> ect appears as a distortion in the real space hexagonal shape of the superlattice cell, resulting from unequal distortion of both the relative length and angular alignment of the reciprocal lattice vectors. Distorted moir´e patterns are commonly observed in HOPG when lattice rotation angles ( GLYPH<18> ) are small. We examine the e GLYPH<11> ect of a uniform strain in the top layer relative to an un- strained layer below. For simplicity, we demonstrate the e GLYPH<11> ect of a relative strain in the arm chair direction on the observed superlattice periodicity. Strain breaks the symmetry of the reciprocal lattice vec- tors, so we now consider the case where the lattice has been uniformly strained along the arm chair direction, such that j a $^{0}$$_{i}$j = (1 + GLYPH<15>$_{i}$ ) j a$_{i}$ j , where i = 0 ; GLYPH<6> . The vector k$_{0}$ along the di- rection of strain does not change angular orientation ( ’$_{0}$ = 0), so under rotation This leads to a change in the wavelength of the interfer- ence pattern along the strain direction as given by the law of k 0 0 = 1 (1 + GLYPH<15>$_{0}$ ) ˆ R ( GLYPH<18> ) k$_{0}$ : (4) D$_{3}$ = D$_{1}$D$_{2}$ q D 2 1 + D 2 2 GLYPH<0> 2 D$_{1}$D$_{2}$ cos ( GLYPH<1> GLYPH<30> ) : (3) FIG. 2: Multibias imaging on the moir´e pattern in Figure 1a. Each image is 5 nm GLYPH<2> 5 nm, and was taken at the same spatial location and tunnel current (100 pA). In (a), only the moir´e pattern of the top two layers is imaged. (b), (c) show the transition as the tip moves closer to the surface. In (d), the tip has pushed in far enough to only image the lower layer moir´e pattern. We believe that by V$_{S}$ = GLYPH<0> 0 : 05 V, the tip is actually in contact with the surface. IV. STRAIN