List of Figures

• Some massive object attached to a wall by a spring. We assume the object is only allowed to move in one dimension, toward or away from the wall.
• A variation on a simple pendulum. Here, not only gravity, but also a spring attached to a wall works to keep the mass in the straight-down position.
• The two ``independent'' motions possible for two pendula, connected by a spring. In (a), the two pendula swing together, and the spring does nothing at all. In (b), the two pendula swing oppositely, and the center of the spring stays fixed; with regard to either pendulum, it could as well be attached to a wall.
• The resting behavior of any object hanging from a spring under the influence of the earth's gravity. How far down the object hangs depends on its mass and on properties of the spring.
• An object on the end of a spring, from the perspective of a speck of dust pinned between the object and the end of the spring. Now both the spring and the large ``object'' are part of the dust's ``external world''.
• A ``time-lapse'' picture of a rock, falling after it has been thrown sideways.
• The path followed by a rock, thrown up and sideways from the ground to some height, and allowed to fall back to Earth.
• Two imaginable paths through space for a falling rock. Part (a) shows the smooth path nature chooses. Part (b) shows a non-catual path, composed of two ``pieced-together'' parts of possible different actual paths.
• The ``path'' occupied by a stretched rubber band, lying on a flat table.
• Some imaginable ``paths'' for a stretched rubber band. (a) is the actual configuration on a real flat table. (b) is a non-actual configuration made from ``pieced-together'' parts of actual different configurations, which could have occurred had the ends been held differently.
• A stretched rubber band, straight except where we force a small ``kink'' into it. The kink could be put anywhere along the length of the band.
• A line of baffles between some light source and the eye. Unless the holes in all of the baffles lie in a suitably striaght line, the eye sees none of the light.
• Light reflecting from a mirror to the eye. A baffle has been added, so that only light that travels ``by way of the mirror'' is allowed to the eye.
• A rubber band, stretched to mimic the ``path'' of light bouncing off a mirror. The finger, here, only allows paths which touch the mirror at some point, but the finger itself can slide back and forth on the mirror, and the rubber is free to distribute itself by sliding beneath the finger as well. Part (a) shows the ``symmetric'' path actually taken by light or the rubber, and part (b) an asymmetric path we could force by holding the finger to one side.
• Paths of light through a focusing lens.
• The sequence of cutting apart the frames of a movie to make a ``flipbook''. (a): The frames as parts of the original movie. (b): cutting frames to make a stack. (c): The stack shown edge-on for simplicity. This is an example of a space-time diagram.
• A space-time picture of a rock thrown straight up and allowed to fall back to Earth. Now time is indicated across the page.
• Space-time diagrams of actual imaginable paths for a thrown rock. The various paths correspond to throwing the rock to various heights.
• Three (somewhat useless) ``time-lapse'' pictures of the trajectory of a mass hanging from a spring, as it traces out a bouncing motion in space. (a): The mass just sits there. (b): It bounces straight up and down a small amount. (c): It bounces straight up or down a larger amount. In all three cases, it simply re-traces its path over and over again, so the time-lapse pictures are just straight lines.
• A space-time diagram of the same possible motions of a mass hanging from a spring, as it bounces up and down. The different paths correspond to different ``greatest extensions'' of travel from its resting point. (Note the similarity to the possible paths of light through a focusing lens.
• Some imaginable (but not actual!) paths for a falling object. We have exercised restraint, and at least considered paths that only go ``forward'' in time.
• One imaginable path, modified by a small bump to produce another imaginable path. Like the kink in a rubber band, this bump could be anywhere on the path.
• A space-time diagram of imaginable motions for a falling rock. Again, part (a) is an actual motion, and part (b) represents a non-actual motion made by piecing together parts of different actual paths.
• Two paths, which differ from each other only by an overall shift in some direction which is a symmetry.
• An imagined path for a falling rock, composed of two pieces of different possible actual paths.
• Space-time diagram of the actual, ``straight-line'' path for the motion of some object.
• The actual straight path, and a slightly different path, obtained from it by slightly speeding up (and thus moving further ahead) at one moment, and then slowing down (to return to the previous path) at a later moment.
• Two possible shorter ``pieces of an actual path'', concatenated to make a real, longer actual path.
• A small piece of a path, over which nothing important changes. Both the time interval and the little bit of action can be divided into two smaller pieces, arbitrarily. Therefore, they must be proportional.
• Two possibilities for the slope at the bottom of a valley. (a): The valley is smooth, and the slope is zero. (b): The valley is sharp, and people on different sides disagree on even what the slope at the bottom is.
• A plot of the absolute value of real numbers. The number lies at the bottom of a ``valley'' of this function, which is not smooth.
• A diagram of the ``idea structure'' built on the notion of the field.
• The process of shifting sideways the little sphere that we use to compute ``averages'' when predicting the evolution of a field. The new center, whose evolution is desired, was one of the boundary places in the previous computation.
• A model for how pressure imbalances cause air to move. The region of interest could be drawn inside a small, imaginary ``box'', and the pressures around it push on the walls of the box, to make it move. The box itself is imaginary, so the only mass that resists motion is the mass of the air inside.
• A small mass of air, originally set in motion by the pressures around it, can ``overshoot'', compressing the air in front and decompressing it behind.
• Pictures of the rules that determine how fluids move. (a): Rule for motion in response to pressure. (b): Rule for expansion and contraction. (c): Rule for Pressure in relation to volume.
• Pictures of height-waves on water and sound-pressure-waves in air.
• The heights of a water surface, when two ripples are present. Note how the height disturbance at each place is the sum of the disturbances from each ripple separately.
• A double-slit experiment done with water waves. The first slit is used to ensure that a smooth wave arrives at the two later slits. The ``screen'' here is just some place where we choose to measure the heights. A real wall is not used, because reflection of the real water waves would greatly complicate the pattern.
• A general wave into which one might deform a stretched string, before letting it go.
• The first three fundamental modes that can be excited in a stretched string. (a): Mode one. (b): Mode two. (c): Mode three.
• The process of adding modes to a string. (a): The first mode only. (b): The first mode minus some of the second. (c): plus some of the third
• Fundamental modes on a stretched membrane. These must now be labeled by their behavior in each direction. (a): The one-one mode on a membrane. (b): The one-two mode on a membrane. (c): The one-three mode on a membrane.
• Two symmetric modes on a stretched, square membrane. (a): The four-two mode on a membrane. (b): The two-four mode on a membrane.
• A graph of the number of fundamental modes for light in a box.
• The flow field for water emerging from the mouth of a pipe into a reservoir.
• Assigning a single arrow to describe the ``amount'' of circulation in a flow.
• The arrows along the edge of a circle add to zero, whenever the flow has no circulation ``in the direction'' of that circle.
• The ``right-hand-rule'' always assigns the same relation to flow around a circle (as indicated by the fingers) and the direction along the axis (as indicated by the thumb), no matter how the hand is oriented.
• An alternative way of adding arrows, this time around a square circumference, to assign a ``magnitude'' to a circulation.
• The divergence of a circulation is zero. Each edge of each face on the cube samples the same part of the flow field as the edge it shares with another face, but in the opposite direction. Therefore every arrow appears twice in the total sum, once positively and once negatively.