Supplementary Information In this section, a detailed study is presented on how the on-axis properties change according to the values of the geometric parameters li as well as i where (i=1,2). Starting with the transition in the Poisson’s ratios i 3 and 3i i 1, 2 from positive to negative values as 1 or 2 decrease and pass through the value of zero, we note that whilst the on-axis Poisson’s ratios in the Ox1 Ox3 and Ox2 Ox3 planes for loading in Ox3 direction change from positive to negative (or vice-versa) as changes from positive to negative values in a continuous and smooth manner, this is not the case for the Poisson’s ratios in the same planes when loading in the Ox1 or Ox2 direction where the change in sign of the Poisson’s ratio occurs in a discontinuous asymptotic manner. This behaviour may be physically explained by the fact that when the structure is pulled along the aforementioned directions, the system locks when the angle becomes zero, in analogy to what happens in the 2D model (Evans et al. 1994). This means that further pulling in that direction will not result in any further deformation of the structure, hence it would not be possible to change the angle i from a negative value to a positive value simply by pulling in the Oxi direction and similarly to change the system from a positive value of i to a negative value. This behaviour is also accompanied by a tendency of the Young’s moduli Ei to approach infinity as the angles i approach zero, indicating locking of the structure. From a mathematical point of view, this asymptotic behaviour can be explained by means of the equations (22-27) obtained in the analytical model for Ei and i 3 for i 1, 2 which simplify to: Ei X i2 d [ i ] 1 4 k h 2 i V li sin 2 i d i (A1) 1 i3 d 3 i i d i X i cos i X 3 sin i X X i cot i (A2) 3 Both of these expressions contain the term d i in the denominator which contains the term sin i . i Due to the presence of the sin i term in the denominator of the equations for 13 and 23 , the Poisson’s ratio varies asymptotically with the angle as i 0 . (The term sin ( i ) is positive when 0 i 2 , negative when 2 i 0 and zero when i 0 .) This means that as the angle approaches 0° (from the left) the Poisson’s ratio tends to -∞, whilst as the angle approaches 0° (from the right) the Poisson’s ratio tends to +∞. Also, this presence of the term sin i in the denominator of i 3 suggests that as the angle i approaches ±90°, the Poisson’s ratio goes to zero. As noted above, this behaviour is also accompanied by the tendency for Ei as i 0 which can be explained by the presence of the term sin 2 i in the denominator of positive apart when i 0 in which case sin 2 i 0 , i.e. i Ei (the term sin 2 i is always goes to zero, independently from which side, Ei tends to infinity). In contrast to all this, the profile of the variation of the Poisson’s ratios in these same Oxi Ox3 planes i 1, 2 for loading in the Ox3 direction with i , is very different since in such cases the change in the sign of the Poisson’s ratio occurs gradually and continuously. This may be explained by the fact that in this case, the change in the sign of the Poisson’s ratio occurs due to the term d i becoming zero, which term is found in the numerator of the expression for 3i and not found in 3 the expression for E3 . Physically this means that one can pull the structure from a negative angle to a positive angle in a one continuous motion without encountering any locking positions. A closer look at these expressions will however suggest that for loading in the Ox3 direction, these systems would still lock upon tensile loading, something which occurs when 1 2 90o . The case when 2 1 2 90o would make the denominator of E3 and that of 31 and 32 become equal to zero. o Here it should be noted that if only one of these angles reaches 90 , then the system would not be locked since deformation can continue through an increase in the angle in the other plane, something which obviously may not be observed in 2D systems. The same would happen when 1 2 90o but this time, locking would occur due to compressive loading. If we now look at how the different parameters l1 , l2 and l3 affect the magnitude of the Young’s moduli and the Poisson’s ratios in the Ox1 Ox3 and Ox3 Ox2 plane, we note that the parameter l3 , i.e. the length of the vertical rib, has an effect such that increasing its value will decrease the value of the Young’s moduli E1 and E2 but increases the value of E3 . This latter effect may, prima facie, appear as surprising as it means that a decrease in density will result in a stiffening of the structure, but this may be easily explained by the fact that irrespective of the value of l3 , the extension of the unit cell dimension in the Ox3 dimension as a result of a stress applied in Ox3 direction is independent of the value of l3 . However, the actual dimension of the un-deformed unit cell will increase with an increase in the value of l3 , hence the resulting strain from the same amount of stress in the Ox3 direction will be larger for systems having small values of l3 compared to systems having large values of l3 . On the other hand, the decrease in the magnitude of the Young’s moduli E1 and E2 on increasing l3 is due to the fact that for a given amount of stress in the Ox1 or Ox2 direction, an increase in l3 will result in a larger cross-sectional area of the unit cell in the orthogonal direction to where the stress is applied with the result that forces acting on each of the ribs cause the hinging deformation to be larger in magnitude, i.e. for a given stress in the Ox1 or Ox2 direction, the extent of hinging increases as l3 increases, and thus E1 and E2 decrease. Such trends in the moduli may also be observed in the 2D model and may be explained using similar arguments. 3 In the case of the Poisson’s ratios, an increase of the parameter l3 will result in a decrease in the magnitude of 13 and 23 and an increase in the magnitude of 31 and 32 . These dependencies of the Poisson’s ratios 13 , 23 , 31 and 32 on l3 may be explained using similar arguments as previously presented. Obviously, there is no effect on the magnitude of the Poisson’s ratios 12 and 21 since these have a constant value of zero which is independent of the geometric parameters used. The relationship between the Young’s moduli and Poisson’s ratios and the values of l1 and l2 is slightly more complex since these parameters will also have an effect on the magnitude of the moment acting on the hinge for a given force acting on the ribs. To explain these effects, one may look at the simplified systems where l1 l2 l and 1 2 , in which case the expressions for the Young’s moduli and Poisson’s ratios (equations (22-30)) simplify to i 1, 2 : Ei 2k h 1 2 2 l3 2l sin l sin i3 l cos Xi cot cot X3 l3 2l sin E3 2kh 3i l3 2l sin l 4 cos3 X 3 tan l3 2l sin tan Xi 2 l cos 2 (A3) (A4) First of all, it may be noted that the simplified expressions suggest that in the case when l1 l2 l and 1 2 , the on-axis Poisson’s ratio in the Ox1 Ox3 and Ox2 Ox3 planes are such that as in the 2D case, 3i1 i 3 . However, in the 3D case 3i1 2 i 3 whilst in the 2D case xy1 yx (i.e. there is a factor of two which results from the 3D geometry). Furthermore, these simplified equations clearly highlight the fact that an increase in the value of l results in a decrease in the value of the Young’s moduli in all three directions. This can be explained by the fact that an increase in the value of l will result in a larger moment being applied, something which is caused both by an increase in the magnitude of the force acting perpendicularly to the rib (there is an increase in the surface area onto which the stress is applied) and also due to the increased 4 leverage effect that results when l increases. In the case of the Poisson’s ratio, for physically realisable structures where l3 0 , an increase in l will result in a change of the aspect ratio which will affect its Poisson’s ratio, as was the case for the 2D model. In fact, an increase in l will result in a bigger relative increase in X i i 1, 2 than in X 3 with the result that as l increases, i 3 increases whilst 3i decreases. In other words, as l increases, for 13 and 23 the ratio between the numerator and the denominator of the equation increases resulting in an increase of both Poisson’s ratios whereas for the Poisson’s ratios 31 and 32 , the reverse applies; an increase in l results in a decrease in the Poisson’s ratio since on increasing l the ratio between the numerator and the denominator of the equation decreases. 5