100 Years of A4: The Story Behind the World's Most Iconic Paper Size
This year marks the 100th anniversary of the A4 paper size standard – a measurement so ubiquitous that we rarely stop to consider its origins or significance.
Yet this humble rectangle of paper has a fascinating history and some clever mathematics behind its dimensions.
Let's explore the story of A4 and why it has become the world's most widely used paper size.
The Birth of the A Series
The A series of paper sizes was first standardised in Germany in 1922 as DIN 476. However, the concept dates back even further to 1786, when German scientist Georg Christoph Lichtenberg first proposed the idea of a paper size with a 1:√2 aspect ratio.
This ratio is key to understanding what makes A4 special. Each size in the A series is exactly half the area of the previous size, while maintaining the same proportions. So A4 (210 x 297mm) is half of A3, which is half of A2, and so on, up to A0, which has an area of exactly one square meter.
A4's Global Adoption
While Germany pioneered the A series standard, it took several decades for it to spread globally. The International Organization for Standardization (ISO) adopted it in 1975, helping to drive worldwide acceptance.
Today, A4 is the standard paper size in nearly every country, with only a few holdouts, such as the United States and Canada, still primarily using letter size (216 x 279mm).
Why A4 Works So Well
The genius of the A series lies in its proportions. The 1:√2 ratio means that when you fold an A4 sheet in half, you get two A5 sheets with exactly the same proportions.
This makes it incredibly easy to scale documents up or down between sizes without distortion.
Some key benefits of A4 include:
Easy scaling between sizes (just multiply or divide dimensions by √2)
Efficient use of paper – minimal waste when cutting larger sheets
Standardisation makes filing, binding and storage simpler
Works well for both print and digital documents
A4 at Springfield Papers
At Springfield Papers, we've been supplying A4 paper to businesses, schools and organisations for over 40 years.
Whether you need bright white copier paper, coloured paper for presentations, or specialty papers for art projects, we stock a wide range of high-quality A4 options to suit every use.
Our expert team can advise on the best paper choices for any application.
Looking to the Future
As we celebrate 100 years of A4, it's worth considering what the next century might hold.
While digital technology has reduced paper usage in some areas, physical documents remain fundamental in many fields. The A4 standard seems likely to endure, continuing to provide a universal format that bridges the physical and digital worlds.
At Springfield Papers, we're committed to sustainable paper sourcing and production. We work with mills and brands that prioritise responsible forestry practices to ensure A4 paper will be available for generations to come.
Whether you need a single ream or a pallet load, we're here to supply all your A4 paper needs. Get in touch with our team today to discuss your requirements or place an order. Here's to another century of this brilliantly designed paper standard!
Single knots at 1/3 and 2/3 establish a spline of three cubic polynomials meeting with C2parametric continuity. Triple knots at both ends of the interval ensure that the curve interpolates the end points
In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial (parametric) curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design.
The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes.
Introduction
The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data may be either one-dimensional or multi-dimensional. Spline functions for interpolation are normally determined as the minimizers of suitable measures of roughness (for example integral squared curvature) subject to the interpolation constraints. Smoothing splines may be viewed as generalizations of interpolation splines where the functions are determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure. For a number of meaningful definitions of the roughness measure, the spline functions are found to be finite dimensional in nature, which is the primary reason for their utility in computations and representation. For the rest of this section, we focus entirely on one-dimensional, polynomial splines and use the term "spline" in this restricted sense.
History
According to Gerald Farin, B-splines were explored as early as the nineteenth century by Nikolai Lobachevsky at Kazan University in Russia.[1]
Before computers were used, numerical calculations were done by hand. Although piecewise-defined functions like the sign function or step function were used, polynomials were generally preferred because they were easier to work with. Through the advent of computers, splines have gained importance. They were first used as a replacement for polynomials in interpolation, then as a tool to construct smooth and flexible shapes in computer graphics.
It is commonly accepted that the first mathematical reference to splines is the 1946 paper by Schoenberg, which is probably the first place that the word "spline" is used in connection with smooth, piecewise polynomial approximation. However, the ideas have their roots in the aircraft and shipbuilding industries. In the foreword to (Bartels et al., 1987), Robin Forrest describes "lofting", a technique used in the British aircraft industry during World War II to construct templates for airplanes by passing thin wooden strips (called "splines") through points laid out on the floor of a large design loft, a technique borrowed from ship-hull design. For years the practice of ship design had employed models to design in the small. The successful design was then plotted on graph paper and the key points of the plot were re-plotted on larger graph paper to full size. The thin wooden strips provided an interpolation of the key points into smooth curves. The strips would be held in place at discrete points (called "ducks" by Forrest; Schoenberg used "dogs" or "rats") and between these points would assume shapes of minimum strain energy. According to Forrest, one possible impetus for a mathematical model for this process was the potential loss of the critical design components for an entire aircraft should the loft be hit by an enemy bomb. This gave rise to "conic lofting", which used conic sections to model the position of the curve between the ducks. Conic lofting was replaced by what we would call splines in the early 1960s based on work by J. C. Ferguson at Boeing and (somewhat later) by M.A. Sabin at British Aircraft Corporation.
The word "spline" was originally an East Anglian dialect word.
Work was also being done at Pratt & Whitney Aircraft, where two of the authors of (Ahlberg et al., 1967) — the first book-length treatment of splines — were employed, and the David Taylor Model Basin, by Feodor Theilheimer. The work at General Motors is detailed nicely in (Birkhoff, 1990) and (Young, 1997). Davis (1997) summarizes some of this material.
We begin by limiting our discussion to polynomials in one variable. In this case, a spline is a piecewisepolynomialfunction. This function, call it S, takes values from an interval [a,b] and maps them to the set of real numbers, We want S to be piecewise defined. To accomplish this, let the interval [a,b] be covered by k ordered, disjoint subintervals,
On each of these k "pieces" of [a,b], we want to define a polynomial, call it Pi. On the ith subinterval of [a,b], S is defined by Pi,
The given k + 1 points ti are called knots. The vector t = (t0, …, tk) is called a knot vector for the spline. If the knots are equidistantly distributed in the interval [a,b] we say the spline is uniform, otherwise we say it is non-uniform.
If the polynomial pieces Pi each have degree at most n, then the spline is said to be of degree≤ n (or of ordern + 1).
If in a neighborhood of ti, then the spline is said to be of smoothness (at least) at ti. That is, at ti the two polynomial pieces Pi–1 and Pi share common derivative values from the derivative of order 0 (the function value) up through the derivative of order ri (in other words, the two adjacent polynomial pieces connect with loss of smoothness of at most n – ri)
A vector r = (r1, …, rk–1) such that the spline has smoothness at ti for i = 1, …, k – 1 is called a smoothness vector for the spline.
Given a knot vector t, a degree n, and a smoothness vector r for t, one can consider the set of all splines of degree ≤ n having knot vector t and smoothness vector r. Equipped with the operation of adding two functions (pointwise addition) and taking real multiples of functions, this set becomes a real vector space. This spline space is commonly denoted by
In the mathematical study of polynomial splines the question of what happens when two knots, say ti and ti+1, are taken to approach one another and become coincident has an easy answer. The polynomial piece Pi(t) disappears, and the pieces Pi−1(t) and Pi+1(t) join with the sum of the smoothness losses for ti and ti+1. That is, where ji = n – ri. This leads to a more general understanding of a knot vector. The continuity loss at any point can be considered to be the result of multiple knots located at that point, and a spline type can be completely characterized by its degree n and its extended knot vector
where ti is repeated ji times for i = 1, …, k – 1.
A parametric curve on the interval [a,b] is a spline curve if both X and Y are spline functions of the same degree with the same extended knot vectors on that interval.
Examples
Suppose the interval [a, b] is [0, 3] and the subintervals are [0, 1], [1, 2], [2, 3]. Suppose the polynomial pieces are to be of degree 2, and the pieces on [0, 1] and [1, 2] must join in value and first derivative (at t = 1) while the pieces on [1, 2] and [2, 3] join simply in value (at t = 2). This would define a type of spline S(t) for which
would be a member of that type, and also
would be a member of that type. (Note: while the polynomial piece 2t is not quadratic, the result is still called a quadratic spline. This demonstrates that the degree of a spline is the maximum degree of its polynomial parts.) The extended knot vector for this type of spline would be (0, 1, 2, 2, 3).
The simplest spline has degree 0. It is also called a step function. The next most simple spline has degree 1. It is also called a linear spline. A closed linear spline (i.e, the first knot and the last are the same) in the plane is just a polygon.
A common spline is the natural cubic spline. A cubic spline has degree 3 with continuity C2, i.e. the values and first and second derivatives are continuous. Natural means that the second derivatives of the spline polynomials are zero at the endpoints of the interval of interpolation.
Thus, the graph of the spline is a straight line outside of the interval, but still smooth.
Algorithm for computing natural cubic splines
Cubic splines are of the form
Given set of coordinates we wish to find set of n splines Si(x) for i = 0, …, n – 1.
These must satisfy:
Let us define one cubic spline S as a 5-tuple (a, b , c, d, xt) where a, b, c, d correspond to coefficients in the form shown earlier and xt is equal to xj.
Algorithm for computing Natural Cubic Splines: Input: set of coordinates C, with |C| = n + 1 Output: set splines which is composed of n 5-tuples.
Create new array a of size n + 1 and for i = 0, …, n set
Create new arrays b and d, each of size n.
Create new array h of size n and for i = 0, …, n – 1 set
Create new array α of size n and for i = 1, …, n – 1 set
Create new arrays c, l, μ, z, each of size n + 1.
Set
For i = 1, …, n – 1 set the following:
Set
For j = n – 1, n – 2, …, 0, set the following:
Create new set Splines and call it output_set. Populate it with n splines S.
For i = 0, …, n – 1 set the following:
Output output_set
Notes
It might be asked what meaning more than n multiple knots in a knot vector have, since this would lead to continuities like at the location of this high multiplicity. By convention, any such situation indicates a simple discontinuity between the two adjacent polynomial pieces. This means that if a knot ti appears more than n + 1 times in an extended knot vector, all instances of it in excess of the (n + 1)th can be removed without changing the character of the spline, since all multiplicities n + 1, n + 2, n + 3, etc. have the same meaning. It is commonly assumed that any knot vector defining any type of spline has been culled in this fashion.
The classical spline type of degree n used in numerical analysis has continuity which means that every two adjacent polynomial pieces meet in their value and first n − 1 derivatives at each knot. The mathematical spline that most closely models the flat spline is a cubic (n = 3), twice continuously differentiable (C2), natural spline, which is a spline of this classical type with additional conditions imposed at endpoints a and b.
Another type of spline that is much used in graphics, for example in drawing programs such as Adobe Illustrator from Adobe Systems, has pieces that are cubic but has continuity only at most This spline type is also used in PostScript as well as in the definition of some computer typographic fonts.
Many computer-aided design systems that are designed for high-end graphics and animation use extended knot vectors, for example Autodesk Maya. Computer-aided design systems often use an extended concept of a spline known as a Nonuniform rational B-spline (NURBS).
If sampled data from a function or a physical object is available, spline interpolation is an approach to creating a spline that approximates that data.
General expression for a C2 interpolating cubic spline
The general expression for the ith C2 interpolating cubic spline at a point x with the natural condition can be found using the formula
where
are the values of the second derivative at the ith knot.
are the values of the function at the ith knot.
Representations and names
For a given interval [a,b] and a given extended knot vector on that interval, the splines of degree n form a vector space. Briefly this means that adding any two splines of a given type produces spline of that given type, and multiplying a spline of a given type by any constant produces a spline of that given type. The dimension of the space containing all splines of a certain type can be counted from the extended knot vector:
The dimension is equal to the sum of the degree plus the multiplicities
If a type of spline has additional linear conditions imposed upon it, then the resulting spline will lie in a subspace. The space of all natural cubic splines, for instance, is a subspace of the space of all cubic C2 splines.
The literature of splines is replete with names for special types of splines. These names have been associated with:
The choices made for representing the spline, for example:
using basis functions for the entire spline (giving us the name B-splines)
The choices made in forming the extended knot vector, for example:
using single knots for Cn–1 continuity and spacing these knots evenly on [a,b] (giving us uniform splines)
using knots with no restriction on spacing (giving us nonuniform splines)
Any special conditions imposed on the spline, for example:
enforcing zero second derivatives at a and b (giving us natural splines)
requiring that given data values be on the spline (giving us interpolating splines)
Often a special name was chosen for a type of spline satisfying two or more of the main items above. For example, the Hermite spline is a spline that is expressed using Hermite polynomials to represent each of the individual polynomial pieces. These are most often used with n = 3; that is, as Cubic Hermite splines. In this degree they may additionally be chosen to be only tangent-continuous (C1); which implies that all interior knots are double. Several methods have been invented to fit such splines to given data points; that is, to make them into interpolating splines, and to do so by estimating plausible tangent values where each two polynomial pieces meet (giving us Cardinal splines, Catmull-Rom splines, and Kochanek-Bartels splines, depending on the method used).
For each of the representations, some means of evaluation must be found so that values of the spline can be produced on demand. For those representations that express each individual polynomial piece Pi(t) in terms of some basis for the degree n polynomials, this is conceptually straightforward:
For a given value of the argument t, find the interval in which it lies
Look up the polynomial basis chosen for that interval
Find the value of each basis polynomial at t:
Look up the coefficients of the linear combination of those basis polynomials that give the spline on that interval c0, ..., ck–2
Add up that linear combination of basis polynomial values to get the value of the spline at t:
For a representation that defines a spline as a linear combination of basis splines, however, something more sophisticated is needed. The de Boor algorithm is an efficient method for evaluating B-splines.
References
Farin, G. E. (2002). Curves and surfaces for CAGD: a practical guide. Morgan Kaufmann. p. 119.
Ferguson, James C, Multi-variable curve interpolation, J. ACM, vol. 11, no. 2, pp. 221-228, Apr. 1964.
Ahlberg, Nielson, and Walsh, The Theory of Splines and Their Applications, 1967.
Birkhoff, Fluid dynamics, reactor computations, and surface representation, in: Steve Nash (ed.), A History of Scientific Computation, 1990.
Bartels, Beatty, and Barsky, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, 1987.
Birkhoff and de Boor, Piecewise polynomial interpolation and approximation, in: H. L. Garabedian (ed.), Proc. General Motors Symposium of 1964, pp. 164–190. Elsevier, New York and Amsterdam, 1965.
Charles K. Chui, Multivariate Splines, SIAM, ISBN 978-0-898712261 (1987).
Ming-Jun Lai , and Larry L. Schumaker, Spline Functions on Triangulations, Cambridge Univ. Press, ISBN 978-0-521-87592-9 (2007).
Stoer & Bulirsch, Introduction to Numerical Analysis. Springer-Verlag. p. 93-106. ISBN 0387904204
Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math., vol. 4, pp. 45–99 and 112–141, 1946.
Young, Garrett Birkhoff and applied mathematics, Notices of the AMS, vol. 44, no. 11, pp. 1446–1449, 1997.
Chapra, Canale, "Numerical Methods for Engineers" 5th edition.
Larry, Schumaker, "Spline Functions: Computational Methods", SIAM, ISBN 978-1-611973-89-1, (2015).
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