# Graph paper

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Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves. It is commonly found in mathematics and engineering education settings and in laboratory notebooks. Graph paper is available either as loose leaf paper or bound in notebooks.

Paper is a thin material produced by pressing together moist fibres of cellulose pulp derived from wood, rags or grasses, and drying them into flexible sheets. It is a versatile material with many uses, including writing, printing, packaging, cleaning, decorating, and a number of industrial and construction processes. Papers are essential in legal or non-legal documentation.

A regular grid is a tessellation of n-dimensional Euclidean space by congruent parallelotopes. Grids of this type appear on graph paper and may be used in finite element analysis, finite volume methods, finite difference methods, and in general for discretization of parameter spaces. Since the derivatives of field variables can be conveniently expressed as finite differences, structured grids mainly appear in finite difference methods. Unstructured grids offer more flexibility than structured grids and hence are very useful in finite element and finite volume methods.

In mathematics, the graph of a function f is, formally, the set of all ordered pairs (x, f ), such that x is in the domain of the function f. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and form thus a subset of this plane, which is a curve in the case of a continuous function. This graphical representation of the function is also called the graph of the function.

## History

The first commercially published "coordinate paper" is usually attributed to Dr. Buxton of England, who patented paper, printed with a rectangular coordinate grid, in 1794. [1] A century later, E. H. Moore, a distinguished mathematician at the University of Chicago, advocated usage of paper with "squared lines" by students of high schools and universities. [2] The 1906 edition of Algebra for Beginners by H. S. Hall and S. R. Knight included a strong statement that "the squared paper should be of good quality and accurately ruled to inches and tenths of an inch. Experience shows that anything on a smaller scale (such as 'millimeter' paper) is practically worthless in the hands of beginners." [3]

The term "graph paper" did not catch on quickly in American usage. A School Arithmetic (1919) by H. S. Hall and F. H. Stevens had a chapter on graphing with "squared paper". Analytic Geometry (1937) by W. A. Wilson and J. A. Tracey used the phrase "coordinate paper". The term "squared paper" remained in British usage for longer; for example it was used in Public School Arithmetic (1961) by W. M. Baker and A. A. Bourne published in London. [3]

## Formats

• Quad paper, sometimes referred to as quadrille paper from French quadrillé, 'small square', [4] is a common form of graph paper with a sparse grid printed in light blue or gray and right to the edge of the paper. In the U.S. and Canada, it often has two, four or five squares to the inch for work not needing too much detail. Metric paper with similarly sparse grid typically has one or two squares per centimeter.
• Engineering paper, or an Engineer's Pad, [5] is traditionally printed on light green or tan translucent paper. It may have four, five or ten squares per inch. The grid lines are printed on the back side of each page and show through faintly to the front side. Each page has an unprinted margin. When photocopied or scanned, the grid lines typically do not show up in the resulting copy, which often gives the work a neat, uncluttered appearance. In the U.S. and Canada, some engineering professors require student homework to be completed on engineering paper. [5] [6]
• Millimeter paper has ten squares per centimeter and is used for technical drawings.
• Hexagonal paper shows regular hexagons instead of squares. These can be used to map geometric tiled or tesselated designs among other uses.
• Isometric graph paper or 3D graph paper is a triangular graph paper which uses a series of three guidelines forming a 60° grid of small triangles. The triangles are arranged in groups of six to make hexagons. The name suggests the use for isometric views or pseudo-three-dimensional views. Among other functions, they can be used in the design of trianglepoint embroidery. It can be used to draw angles accurately.
• Logarithmic paper has rectangles drawn in varying widths corresponding to logarithmic scales for semi-log plots or log-log plots.
• Normal probability paper is another graph paper with rectangles of variable widths. It is designed so that "the graph of the normal distribution function is represented on it by a straight line", i.e. it can be used for a normal probability plot. [7]
• Polar coordinate paper has concentric circles divided into small arcs or 'pie wedges' to allow plotting in polar coordinates.

A photocopier is a machine that makes copies of documents and other visual images onto paper or plastic film quickly and cheaply. Most modern photocopiers use a technology called xerography, a dry process that uses electrostatic charges on a light-sensitive photoreceptor to first attract and then transfer toner particles onto paper in the form of an image. Heat, pressure or a combination of both is then used to fuse the toner onto the paper. Copiers can also use other technologies such as ink jet, but xerography is standard for office copying. Earlier versions included the Gestetner stencil duplicator, invented by David Gestetner in 1881.

An image scanner—often abbreviated to just scanner, although the term is ambiguous out of context —is a device that optically scans images, printed text, handwriting or an object and converts it to a digital image. Commonly used in offices are variations of the desktop flatbed scanner where the document is placed on a glass window for scanning. Hand-held scanners, where the device is moved by hand, have evolved from text scanning "wands" to 3D scanners used for industrial design, reverse engineering, test and measurement, orthotics, gaming and other applications. Mechanically driven scanners that move the document are typically used for large-format documents, where a flatbed design would be impractical.

Homework, or a homework assignment, is a set of tasks assigned to students by their teachers to be completed outside the class. Common homework assignments may include required reading, a writing or typing project, mathematical exercises to be completed, information to be reviewed before a test, or other skills to be practiced.

In general, graphs showing grids are sometimes called Cartesian graphs because the square can be used to map measurements onto a Cartesian (x vs. y) coordinate system. It is also available without lines but with dots at the positions where the lines would intersect.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

## Related Research Articles

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

A logarithmic scale is a nonlinear scale used for a large range of positive multiples of some quantity. Common uses include earthquake strength, sound loudness, light intensity, and pH of solutions.

A chart is a graphical representation of data, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabular numeric data, functions or some kinds of qualitative structure and provides different info.

An engineering drawing, a type of technical drawing, is used to fully and clearly define requirements for engineered items.

In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.

Ruled paper is writing paper printed with lines as a guide for handwriting. The lines often are printed with fine width and in light colour and such paper is sometimes called feint-ruled paper. Additional vertical lines may provide margins or act as tab stops or create a grid for plotting data; for example, graph paper is divided into squares by horizontal and vertical lines.

Grapher is a computer program bundled with macOS since version 10.4 that is able to create 2D and 3D graphs from simple and complex equations. It includes a variety of samples ranging from differential equations to 3D-rendered Toroids and Lorenz attractors. It is also capable of dealing with functions and compositions of them. One can edit the appearance of graphs by changing line colors, adding patterns to rendered surfaces, adding comments, and changing the fonts and styles used to display them. Grapher is able to create animations of graphs by changing constants or rotating them in space.

Many letters of the Latin alphabet, both capital and small, are used in mathematics, science and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, physical entities. Certain letters, when combined with special formatting, take on special meaning.

Point plotting is an elementary mathematical skill required in analytic geometry. Invented by René Descartes and originally used to locate positions on military maps, this skill is now assumed of everyone who wants to locate grid 7A on any map.

In graph theory, the Cartesian productGH of graphs G and H is a graph such that

In science and engineering, a semi-log graph or semi-log plot is a way of visualizing data that are related according to an exponential relationship. One axis is plotted on a logarithmic scale.

An exercise book or composition book is a notebook that is used in schools to copy down schoolwork and notes. A student will usually have a different exercise book for each separate lesson or subject.

In the context of a spatial index, a grid or mesh is a regular tessellation of a manifold or 2-D surface that divides it into a series of contiguous cells, which can then be assigned unique identifiers and used for spatial indexing purposes. A wide variety of such grids have been proposed or are currently in use, including grids based on "square" or "rectangular" cells, triangular grids or meshes, hexagonal grids and grids based on diamond-shaped cells.also "global grid" if it covers the entire surface of the globe)

Two-dimensional space is a geometric setting in which two values are required to determine the position of an element. In mathematics, it is commonly represented by the symbol 2. For a generalization of the concept, see dimension.

An ergograph is a graph that shows a relation between human activities, or agricultural/climate factors, and a seasonal year. The name was coined by Dr. Arthur Geddes of the University of Edinburgh. It can either be a polar coördinate (circular) or a cartesian coördinate (rectangular) graph, and either a line graph or a bar graph.

The following comparison of Adobe Flex charts provides charts classification, compares Flex chart products for different chart type availability and for different visual features like 3D versions of charts.

In mathematics, log-polar coordinates is a coordinate system in two dimensions, where a point is identified by two numbers, one for the logarithm of the distance to a certain point, and one for an angle. Log-polar coordinates are closely connected to polar coordinates, which are usually used to describe domains in the plane with some sort of rotational symmetry. In areas like harmonic and complex analysis, the log-polar coordinates are more canonical than polar coordinates.

## References

1. Friendly, Michael; Denis, Daniel J. (5 July 2006). "Milestones in the history of thematic cartography, statistical graphics, and data visualization" (PDF). York University: 13.
2. Borovik, Alexandre (7 August 2008). "Graphed Paper". Mathematics under the Microscope. Retrieved 25 March 2017.
3. Ballew, Pat. "Some Notes on graph paper". Math Words, and Some Other Words, of Interest. PBallew.net. Retrieved 25 March 2017.
4. "quadrille". Oxford Living Dictionaries. Oxford University Press. 2017.
5. "The Preparation of Engineering Problem Sets" (PDF). Massachusetts Institute of Technology. Technical Communications in Mechanical Engineering. Retrieved 25 March 2017.
6. "Required Homework Format" (PDF). Auburn University. Department of Chemical Engineering. Retrieved 25 March 2017.
7. Prokhorov, A. V. (2011). "Probability graph paper". Encyclopedia of Mathematics. Retrieved 18 January 2014.