In this section, we consider the implications on strength, mass, and other properties when changing lens scale. Here we see two similar triangles. They are similar because the angles a, b, and c are all equal. However, one triangle has side length L and the other has side length 2L. Now we consider the implications on some properties of these triangles. Since the perimeter is simply the sum of the sides, the perimeter of the larger triangle will be double that of the small triangle. So the perimeter will scale linearly with the lens. The area of the triangles, however, scales according to length squared. This motivates the much more general field of scaling analysis, and specifically dynamic scaling and dynamic similarity, which are very useful in robotics. Now instead of triangles, we consider what happens when you scale cubes. As expected, the area of the cubes is proportional to the length squared, and the volume of the cubes is proportional to the length cubed. As you can see, a cube with side length two L, and be thought of as containing eight smaller cubes of side length, L. Suppose you have two ice cubes, one of side length, L, and one of side length, 2L, and they're sitting in the sun. For these ice cubes, the heat that goes into them is proportional to their surface area and the thermal mass of the ice cubes is proportional to their volume. So now scaling will help us answer the question which ice cube will melt first and by what factor. As well as helping us make predictions about what will happen with even larger or smaller ice cubes. Galileo was also very curious about the implications of scaling in animals. He observed that bones of different sizes had different proportions. In this figure, we see one of Galileo's sketches, in which he draws a smaller bone and a larger bone, where the larger bone is disproportionately thicker than the smaller bone. The question he asked is if a larger bone that is three times the length of the smaller bone were to experience the same stresses in a larger animal as compared to a smaller animal. How much disproportionately thicker did the larger bone need to be. Just preserving cross-sectional area as the bone scales does not however tell the whole story. Larger animals typically move relatively more slowly as measured in body length per second compared to smaller animals. Furthermore, larger animals typically adopts a less probe posture than smaller animals. We now consider how a change in posture influences a change in loading of a cartoon of a bone. There are five fundamental types of loading. Compression is a force parallel to the axis of the bone, where the bone is being pushed together. Tension is also parallel to the axis of the bone, but it is being pulled apart. Shear is perpendicular to the axis of the bone as if it is being slid apart. Bending is a result of a force perpendicular to the axis of the bone coming out of the screen that is being twisted. Torsion is the result of a twisting motion, but parallel with the bone. Here we introduce a fundamental notion of stress, which is force normalized by area. Stress is a very important consideration. Depending on how a part is being loaded, there's a certain stress at which it must break.