Harmonograph
Encyclopedia
A harmonograph is a mechanical apparatus that employs pendulum
s to create a geometric image. The drawings created typically are Lissajous curve
s, or related drawings of greater complexity. The devices, which began to appear in the mid-19th century and peaked in popularity in the 1890s, cannot be conclusively attributed to a single person, although Hugh Blackburn, a professor of mathematics
at the University of Glasgow
, is commonly believed to be the official inventor.
A simple, so-called "lateral" harmonograph uses two pendulums to control the movement of a pen
relative to a drawing surface
. One pendulum moves the pen back and forth along one axis and the other pendulum moves the drawing surface back and forth along a perpendicular
axis. By varying the frequency
and phase of the pendulums relative to one another, different patterns are created. Even a simple harmonograph as described can create ellipse
s, spiral
s, figure eights
and other Lissajous figure
s.
More complex harmonographs incorporate three or more pendulums or linked pendulums together (for example hanging one pendulum off another), or involve rotary motion in which one or more pendulums is mounted on gimbal
s to allow movement in any direction.
A particular type of harmonograph, a pintograph, is based on the relative motion of two rotating disks, as illustrated in the links below.
in which
represents frequency, 
represent phase, 
represent amplitude, 
represents damping and 
represents time. If that pendulum can move about two axes (in a circular or elliptical shape), due to the principle of superposition, the motion of a rod connected to the bottom of the pendulum along one axes will be described by the equation
A typical harmonograph has two pendulums that move in such a fashion, and a pen that is moved by two perpendicular rods connected to these pendulums. Therefore the path of the harmonograph figure is described by the parametric equations
An appropriate computer program can translate those equations into a graph that emulates a harmonograph. Applying the first equation a second time to each equation can emulate a moving piece of paper (see the figure below).
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
s to create a geometric image. The drawings created typically are Lissajous curve
Lissajous curve
In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equationswhich describe complex harmonic motion...
s, or related drawings of greater complexity. The devices, which began to appear in the mid-19th century and peaked in popularity in the 1890s, cannot be conclusively attributed to a single person, although Hugh Blackburn, a professor of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at the University of Glasgow
University of Glasgow
The University of Glasgow is the fourth-oldest university in the English-speaking world and one of Scotland's four ancient universities. Located in Glasgow, the university was founded in 1451 and is presently one of seventeen British higher education institutions ranked amongst the top 100 of the...
, is commonly believed to be the official inventor.
A simple, so-called "lateral" harmonograph uses two pendulums to control the movement of a pen
Pen
A pen is a device used to apply ink to a surface, usually paper, for writing or drawing. Historically, reed pens, quill pens, and dip pens were used, with a nib of some sort to be dipped in the ink. Ruling pens allow precise adjustment of line width, and still find a few specialized uses, but...
relative to a drawing surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
. One pendulum moves the pen back and forth along one axis and the other pendulum moves the drawing surface back and forth along a perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
axis. By varying the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
and phase of the pendulums relative to one another, different patterns are created. Even a simple harmonograph as described can create ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
s, spiral
Spiral
In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.-Spiral or helix:...
s, figure eights
Figure 8
Figure 8 may refer to:* 8 , in Arabic numeralsIn entertainment* Figure 8 , by American singer-songwriter Elliott Smith* "Figure of Eight" from the album Flowers in the Dirt by Paul McCartney...
and other Lissajous figure
Lissajous curve
In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equationswhich describe complex harmonic motion...
s.
More complex harmonographs incorporate three or more pendulums or linked pendulums together (for example hanging one pendulum off another), or involve rotary motion in which one or more pendulums is mounted on gimbal
Gimbal
A gimbal is a pivoted support that allows the rotation of an object about a single axis. A set of two gimbals, one mounted on the other with pivot axes orthogonal, may be used to allow an object mounted on the innermost gimbal to remain immobile regardless of the motion of its support...
s to allow movement in any direction.
A particular type of harmonograph, a pintograph, is based on the relative motion of two rotating disks, as illustrated in the links below.
Computer generated harmonograph figure
A harmonograph creates its figures using the movements of damped pendulums. The movement of a damped pendulum is described by the equationin which
represents frequency, represent phase, represent amplitude, represents damping and represents time. If that pendulum can move about two axes (in a circular or elliptical shape), due to the principle of superposition, the motion of a rod connected to the bottom of the pendulum along one axes will be described by the equationA typical harmonograph has two pendulums that move in such a fashion, and a pen that is moved by two perpendicular rods connected to these pendulums. Therefore the path of the harmonograph figure is described by the parametric equations
An appropriate computer program can translate those equations into a graph that emulates a harmonograph. Applying the first equation a second time to each equation can emulate a moving piece of paper (see the figure below).