A Harmonograph is a mechanical instrument consisting of at least two clock-like pendulums to which a pen is attached. A paper is used to draw the harmonic curve when the two pendulums are swinging. Some harmonographs also have a rotating or swinging paper to add more complexity (or creativity) to the output.

A Pantograph does not exist in this field of harmonics drawing. It's the name of what we have given to this instrument. It resembles the harmonograph a bit and instead of two swinging pendulums, two rotating disks are attached to a pencil via two rods, connected like a well known train pantograph, which in turn is drawing its path on a piece of paper. In some cases also an smaller ('inner') disc rotates on a larger ('outer') disk, to allow for a little bit more rotation variations. On the internet alternative names are found, such as Pintograph or Rotating Harmonograph.

The two pictures below may explain better that what we described in words above.

Left: a harmonograph; Right: a pintograph. (Disclaimer: pictures are only used for explanation, no infringement of the owner's rights is intended).

The Harmonograph

There are a number of controls for the harmonograph. The two pendulums are called 'Left' and 'Right ', but they could be called any name (like 1 and 2 or A and B). The paper movement naming can be somewhat confusing, since it's called 'Horizontal' and 'Vertical'. It is basically a rotation where the paper pendulum can swing in a horizontal and vertical direction, showing in real life as a rotating ('ellipsoidal') movement.

Figure above: a photoshopped version of the original (www.karlsims.com) to effectively show a couple of harmonograph variables in the Pattaizer application.

There are 4 identical variables for each pendulum movement. Some are also explained in the picture below. These are (see picture):

• Frequency: the amount of swings per second. In most cases these values are very small.
• Amplitude: the initial amount of distance when starting the pendulum (see picture).
• Damping: a degree of slowing down the pendulum with each complete swing. It's also a measure of the amount of friction of the mechanical hinges that eventually slows down the pendulum until it stops completely. The damping should not be too small to avoid that the pen hardly changes course after a full swing.
• Phase: this is a measure of the timing difference between two pendulums. It seems a bit odd to have a phase for each pendulum while it only matters when multiple pendulums exist. However for the mathematical simulation it certainly adds a flavour to the resulting drawing.

Next to these 4 basic variables also interaction variables are used to define the dependencies between the two pendulums and the (rotating) paper. In the picture below these are shown. The 'L' is the left pendulum, the 'R' stand for the right pendulum, the 'Hor' and 'Ver' apply to the paper ('write disk'). The following interaction parameters can be set:

• Angle: the angle between the direction of the pendulum and the direction of the paper (see picture). The top value of the Angle is the angle between the two pendulums, the bottom value of the angle between the two virtual movements of the write disk. If the latter angle is 90 degrees it will follow a circular movement.
• Rotation: the paper can also rotate (next to an ellipsoid movement due to the 'horizontal' and 'vertical' swings). See picture below. In real life this is mechanically challenging, but in mathematical world we can add all kinds of interesting new variables in the harmonograph.
• Sync pendulums: to avoid settings identical values for the two pendulums you can also check this box. It will also hide one pendulum tab (the right pendulum).
• Sync write disk: similar as the synchronization of the two pendulums the movement of the paper disk can be synced. In this case the paper disk will make a perfect circle when synced.
• The other boxes for 'L', 'R', 'Ver' and 'Hor' can be unchecked if you want that part of the mechanism not to swing.

Note: if you check both sync options you will not get a straight line because the harmonograph does not change directions of pen and paper disk (relative to each other).

• Number of points: the number of lines to draw the Lissajous figure. Usually 1000 lines are enough for an initial evaluation. In some cases you need (or want) to increase the number to make the result look more 'filled'.
• Start offset: in some cases the first line is clearly visible, e.g. standing out on the outside of the drawing. In that case you can try to obscure this starting line by skipping a few lines at start. In most cases the starting point will visually disappear.
• Density: the amount of lines in the Lissajous result while keeping the shape and size. A large value of density may lead to less sharp curves. This options is only available for the Harmonograph. The option that is available for the Pantograph and not for the Harmonograph is the 'Preset' value. Here you can choose from a couple of great presets, showing the possibilities of the Pantograph pattern.
• Note: the pen thickness control is the same as for the other patterns, where the 'Line' value positioned at the top of the interface, also sets the pen width/thickness of the harmonic patterns.
• Autosave current settings: when you have found a nice set of parameters you would like to either save it as a project or keep these settings, so it starts with that great initial picture when the application starts.
• Print settings on canvas: in some cases you want to record the actual settings when you create a High Res bitmap. It can be very hard to recreate that great result again if you forgot to record the used settings. The parameters are printed in the left corner at the bottom of the bitmap.
• Anti-alias output: the drawn lines are sharp lines. To avoid this 'edgy' look of the lines you can activate this option. For very large bitmap outputs this may cause instabilities because it requires approximately 25% more RAM to create the anti-alias.
• Accuracy: the accuracy of anti-aliasing. This value is also directly related to the size of the output bitmap! On some systems the amount of required RAM may not be enough. The value of accuracy will set the amount of RAM needed to anti-alias the resulting bitmap; low values are usually fine but less anti-aliasing, high values might lead to RAM problems on some low-end systems and laptops.
• Do not interpolate points: the High Res output will use approximately 8 times more points between the draft Lissajous picture. For the Harmonograph this is not a real issue, but for the Pantograph this option might be activated to maintain the draft result because of the instability of some settings when enlarged to High Res.

• Dual colour: these two rectangles can be clicked on, defining the colour of the line (left square) and the colour of the background/paper (right square). The 'Boring default' is dark gray pen on a white paper background.
• Colour preset : in case you want to vary your output a couple of examples are available in the application. These colour combinations are selected in the drop-down box (here the 'Boring default' is shown). When you want to add your great new colour combination, press the little button left of this selection box. It will ask for a name and it will add this to the drop-down box next time you start the application.
• Bezier, Line and Multi: there are three ways to draw a line in the Lissajous figure. 'Line' is just connecting the two points which are calculated in the mathematics of the harmono- and pantograph. 'Bezier' is the mathematical term for a line that takes the location of the next point into account adapts its curvature and direction accordingly. This way a Bezier line seems to seamlessly follow a smooth path between the calculated points. It's also the default at start. 'Multi' allows for a change of colours. The colours can be set in several ways:
  • Use a preset from the drop-down list (default is 'Random colours').
  • Select a 'Start colour' and 'End colour' for each preset. The colours can be set using the two colour controls. Each control has a square area for selecting the lightness of the colour and the saturation; and a hue selector on the right. Use the mouse to move the little gray square and line in these controls.
• (No illustration): at the bottom of the interface you'll find two more controls that work for both patterns: 'Paper rotation' and 'Fill-out'. The first will rotate the pattern in view the second will maximize the pattern to fill in the available space. In that case the zoom and pan features won't be active.

Make a  High Resolution picture

The last common parameter is actually the 'Update mosaic' button. When the Pantograph or Harmonograph pattern is selected the button changes it caption to 'HiRes picture'. Both harmonic patterns are automatically updated (the 'Auto-update pattern' does not work here), and to create a High Resolution picture you need to press this button.

Please be aware that text images, as in-app generated, will not be recognized by the two harmonic patterns. To make a text pattern, and then apply a harmonic pattern you first need to create the text image yourself, save as a bitmap, open it in Pattaizer and then use it for the colours of the harmonics.

The Pantograph

The pantograph is more difficult to manage than the harmonograph. One of the reasons is that the pantograph does not easily follow a smooth Lissajous pattern where a repeatedly drawn figure shifts gradually, e.g. by paper rotation or damping. It is based on two stable rotating disks that are both connected to a rod, which come together in a single hinging point, where also the drawing pen is attached. Since the movement damping of the drawing pen is less controlled. The drawing really gets its good looks from a balanced set of parameters that apparently create a slightly varying path once a full cycle of the pen movement has been completed. In many cases the result is absolutely different. But not necessarily bad. Check out these two example in the next picture.

Left: a reasonable good pattern with balanced parameters. Middle: it looks like chaos with no repeating but with a slightly shifting pattern. Right: same as the middle, but now as a High Res rendering on a black paper in full colour. Now it looks like very nice fireworks.

To show how the pantograph works we have compiled a schematic drawing, see below for explanation and definitions.

A number of parameters can be set for the pantograph figure. There are three groups of settings, and two groups of common controls that are the same for the harmonograph (see paragraph above). The settings for the pantograph are:

• Outer diameter: the diameter of the large disk on which a smaller disk is placed, on which the rod is connected (see schematic drawing above).
• Inner diameter: the diameter of the inner disk. This is basically the same as the radius of the connection point of the rod. The inner disk actually does not move relative to the outer disk.

• Distance: the distance between the two large disks (see schematic below).
• Rod length: the length of the two identical rods that connect the inner disk with the pantograph part, and finally attached to the drawing pencil. (see schematic).
• Zoom factor: to enlarge or make the pattern smaller (a part of the pattern can move outside the view). Is not applied in case the pattern is filled-out.
• Pan left/right: this pattern typically tends towards the right, so a bit panning migth bring the pattern back in teh center of the view. is not applied in case the pattern if filled-out.

• Disk (rotation): the rotational speed of the outer disk
• Off-set: the rotational speed difference between the two outer disks. This is a simple way to set the rotation speed for the two disk, then apply a kind of 'brake' to one of the disks, to effectively create a deviation from a repeatable drawing path. It creates extremely beautiful drawings when set to small values. At high values there may not be a sufficiently repeated shape as the basis for the Lissajous from the pantograph.
• Paper: the rotational speed of the paper. As with the 'Off-set' this value can create highly unstable Lissajous patterns. But is also a basis to create extremely unique patterns.
• Preset: this is a quick way to start with a stable set of values. These pre-sets are defined by the application and cannot be changed (well... you can but be careful with editing the 'ColorPresets.dat' file, always make a backup first).

The Video feature of the harmonics

The two harmonic patterns also have a dedicated video settings tab:

The controls are identical to those of the mesh/grid-based patterns, see here for detailed descriptions of each of the script creation and script loading options. There is however a special extra video option available for the two harmonic patterns: 'Animation'. The Animation button will start to record the progress of the actual drawing action of the harmonic pattern. As if you are watching the contraption move and sway while drawing the pattern on a sheet of paper. To record the animation and save as a video file, you also need to check the box 'Record the animation'. If not, the animation will be shown in the GUI, and no frames are saved in a video file.