This and mathematics (particularly geometry). They first

This research essay aims to underline the significant role
mathematics play in typography, specifically in the creation of a typeface, by
defining a connection between typography and mathematics (particularly
geometry). They first part of the essay will consist of a brief history of
typography, underlining the key turning points it as had across time especially
around the sixteenth century. Further on, the next part will focus on
mathematics as a basis for the development of type, in particular the geometric
aspects, such as the golden ratio. A questionnaire will be at the base of the primary research, examining
whether there is any justification for optically correcting mathematical
proportions in typography, and whether doing is so is noticeable to the common


Designers and artists have been long fascinated by geometry. Contemporary
typographers view typography as a craft with a very long history, tracing its
origins back to the first dyes and punches used to make seals and currency in
ancient times. These basic elements of typography are as old as civilization
and the earliest writing systems, however a series of key developments were
eventually drawn together into a singular systematic craft. (Steinberg, 1961)

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Renaissance artists’ and scholars’ fascination with mathematics
included the domain of the alphabet, and they meticulously studied the styles
and proportions of engraved capitals still visible on classical Roman
monuments. (Loxley, 2006) The
development of geometric perspective in the 1420s convinced artists of the
period that the whole of reality could be expressed in numerical formulae. At
the same time, the works of Greek mathematicians such as Pythagoras were deemed
to be the high point of ancient civilization (whose perfection arose from the
systematic, rational application of the laws of geometry). (Guthrie, 1988)

The concept of defining letters mathematically is by
no means new, as it goes back to the fifteenth century and it became rather
highly developed in the early part of the sixteenth century. In a time where
Renaissance men combined mathematics with the real world, a particular interest
in constructing majuscules with rulers and compasses arose. There are numerous
influential books published at the beginning of the sixteenth century, authors
varying from mathematicians to calligraphers to typographers. Some of the most
notable include Albrecht Dürer, Francesco Torniello da Novara, Geoffroy Tory.

Luca de Pacioli, an Italian mathematician, studies the proportions of classical
Roman letters in his book De divina
proportione written in Milan in 1496-98. This was the first printed book
known to describe the Roman alphabet with the help of Leonardo da Vinci’s
drawings. (Knuth, 1979)


The years between the mid-15th century and the early 18th
century proved to be a time of many changes and developments in the world of
typography, with the development of the printing press greatly influencing the
development of full typefaces and their production. (Eisenstein, 1996) In the 15th century, Albrecht Dürer and Francesco Cresci tried to
rationalize letterforms in accordance with geometric principles from a
lettering copy-book. It was not until the twentieth century, however, that
geometry began to influence type in a direct way.



Mathematical typesetting has always challenged the printer:

authors should be very exact in their Copy, and Compositors as careful in
following it, that no alterations may ensue after it is composed; since
changing and altering work of this nature is more troublesome to a Compositor
than can be imagined by one that has no tolerable knowledge of Printing. Hence
it is, that very few Compositors are fond of Algebra, and rather chuse to be
employedon plain work, tho’ less profitable to them than the former; because it
is disagreeable and injures the habit of an expeditious Compositor.” (Smith, 1998)



In 1967, Edward Wright used geometry for his architectural
lettering for the Metropolitan Police headquarter building in London. Related
in part to the Bauhaus idea, geometry was the basis for Paul Renner’s Futura of
1927-9, as pattern drawings from the Bauer Typefoundry.


Nationality does not easily apply to type and culture, as Yvonne
Schwemer-Scheddin has stated: “The concept of ‘nation’ is political, whereas
script is connected directly to language and its geographic linguistic areas.” Languages,
scripts, and typefaces which have represented centuries were intrinsically
related to each other. Earlier Roman capitals became the typographic system
dedicated to monumental inscriptions during the Roman Empire, and later on the
Carolingian minuscule was adopted as the official script for all of the Charlemagne
Empire. In Ireland, as can be seen in books, the Semi Uncial was in use. During
the early years of civilization, unlike in our globalized world, there was no
easy way of communication between various parts of the world.


The idea of defining letters mathematically is in no way new, it
goes back to the fifteenth century and it became rather highly developed in the
early part of the sixteenth century. In a time where Renaissance men combined
mathematics with the real world, there was a particular interest in
constructing majuscules with rulers and compasses. The first person to do so is
thought to be Felice Feliciano, around 1460, whose handwritten manuscript in
the Vatican Library was published 500 years later. Feliciano was an excellent
calligrapher who wanted to a mathematical foundation applied to the principles
of calligraphy. Several other fifteenth century authors made similar
experiments, however the most notable work of this kind appeared in the early
sixteenth century.

Italian mathematician Luca Pacioli, who had previously written the
most influential book on algebra at the time, also published De Divina Proportione (1509), a book
about geometry and the golden section, which included an appendix about
alphabets. Francesco Torniello da Novara published another notable Italian work
on the subject in 1517 titled Opera del
modo de fare le littere maiuscole antique, which focused on enriching, both
geometrically and calligraphically, previously existent fonts. His works
generally focused on adopting the Latin alphabet inscriptions as originally as
possible, whilst improving their geometric conditions. These fonts were not
designed for the printing press, but as a model for artistic inscriptions. He
was also the first to define the ‘point’ as the measure in typography. Similar work appeared in France and
Germany; The German book was probably the most famous and influential: Albrecht
Dürer’s Underweysung der Messung (1525), an
instructional manual in geometry for Renaissance painters. Another rather
popular book was Geoffroy Tory’s Champfleury
(1529), which set a standard for French publishing that in many ways is
still followed today. Nobody carried this work further to minuscule, numerals,
or italic letters and other symbols, until 100 years later when Joseph Moxon
made a detailed study of some letters designed in Holland. The ultimate
refinement of this mathematical approach took place shortly afterwards when
Louis XIV of France commissioned the creation of a Royal Alphabet. A commission
of typographers and artists worked on Louis XIV’s project for over ten years,
starting around 1690.


Francesco Torniello da Novara was
a Milanese typographer who became known for applying geometric specifications
to Latin majuscule fonts. Torniello’s works concentrated on adopting the Latin
alphabet inscriptions as originally as possible, while also improving their
geometric conditions. These fonts were not conceived for the printing press,
but as a model for artistic inscriptions. He designed an 18×18 which served as
a coordinate system for his geometric fonts, which were designed for printing
press usage. In his book Opera del modo da
fare le littere majuscole antique, published in 1517, Torniello improved,
both geometrically and calligraphically, pre-existing fonts. Geometrical
conditions were added to the letters “M”, “R”, “S”, and “T” – in order to
broaden the usage of his fonts in non-Latin texts, he also added the letter
“Z”. He was the first typographer to define the ‘point’ as unit of measure in

Thanks to the reverential and
detailed reference Torniello made in his work regarding Luca Pacioli, it is
safe to assume they must have either met, or Torniello had access to his work Divina Proportione. Torniello’s work
contains geometric representations of all Latin majuscules. Pacioli’s typeface
at this time represented the culmination of development in the Renaissance, which
strove to adopt the Latin letters of ancient inscription and tombstones as
faithfully as possible. This development began in 1460 with a Vatican archived
manuscript by Felice Feliciano, and was continued by Damianus Moyllus (or
Damiano da Moyle?) in either 1483 or 1484. As tools for the
design of individual letters, only rulers and compasses were available.

However, up to and including Pacioli’s work, only the finished designs were
shown along with some design elements such as circles used and the surrounding
rectangle or square. Furthermore, a complete written specification matching the
charts was missing. In this regard, Torniello’s work broke new ground. This
begins with Torniello’s grid that served as the coordinate system. The double
side length of a grid was defined as a ‘point’. Thus, this became the first
known definition of a point as a typographic unit of measure. However, it is
unlikely that Pierre Simon Fournier knew the work of Torniello when he also
introduced the point in 1737 as a typographic unit. The point as a unit of
measure and the grid greatly facilitated Torniello’s specifications in writing.


While Torniello largely took over
the designs of Pacioli, he deviated significantly from his model for the
letters “M”. “R”, “S”, and “T”. In particular, the letters “S” and “T” were
enriched by calligraphic details, which were also occupied by later
typographers. For example, the design of the parallel diagonally cut upper
serif of the “T” in the alphabet by Albrecht Dürer, can be found again. Only
the “M” was considered less successful in the comparative analysis of Giovanni
Mardersteig. Furthermore, Torniello added “Z” to his alphabet, which can be
taken as evidence that these fonts have also been increasingly used for
non-Latin texts. By the letters “R” and “S”, other peculiarities of Torniello’s
construction drawings can also be recognized. Thus, he drew for each circular
arc used in each case the complete circle segment together with the indication
of the respective radius, which was measured in points.


Torniello’s work and the alphabets of his predecessors,
especially those of Pacioli and Sigismondo Fanti (1514), significantly
influenced the further development. Giovambaptista Verini published in 1526 in
the third volume of his work Luminario
an alphabet that derived largely from the designs of Pacioli and Fanti, however
Torniello took over the definition of the point as the ninth of the page
length. In contrast to Pacioli and Torniello, the much larger circles in the
serifs are noticeable, so that they appear much heavier.


In 1528 Albrecht Dürer published his designs for his
alphabet, which he had made in 1525, in Nuremberg. Dürer knew his Italian
predecessors quite well and was guided mainly by Pacioli and Torniello. He was
the first to present both variants, the completed letter based on Pacioli and
the construction sketched in the same way as Torniello’s model, side by side.

Dürer took over the screening from Torniello as well. The sizes of the serifs are,
however, more similar to those of Pacioli. Though they are slightly less
graceful than Torniello’s, they are also much smaller than Verini’s. Shortly
afterwards, but in full knowledge of the work of Albrecht Dürer and the Italian
role models, Geoffroy Tory published his work Champfleury in Paris in 1529. He took over from Albrecht Dürer the
dual representation according to Torniello and Pacioli and also used a grid,
which is dimensioned in contrast to that of Torniello to 10×10. For the serifs, however,
much larger circles were used, which are more alike to those of Verini than the
smaller circles of Torniello or Pacioli. The departure of the circle as a tool
of typography was initiated by Ludovico degli Arrighi in 1523 and in particular
by Francesco Cresci in his work Essemplare
di più sorti lettere of
1560. Cresci re-studied the ancient inscriptions and came to the conclusion
that compass-based constructions do not do justice to the originals. For
example, the adjacent “B” is based on the corresponding form in the inscription
on the pedestal of Trajan’s Column in Rome. Cresci concludes that “in drawing
every curve of each letter they make more circles than a sphere for the most
part contains. I have come to the conclusion that Euclid, the prince of
Geometry, returned to this world of ours, he would never find that the curves
of the letters could, by any means of circles made with compasses, be
constructed according to the proportion and style of ancient letters.” (Swetz,



type designer – or better, let us start calling him the alphabet designer –
will have to seek his task and his responsibility more than before in the
coordination of the tradition in the development of letterforms with the
practical purpose and the needs of the advanced equipment of today…” (Zapf,


The new challenges have
resurrected the quest for the perfect alphabet with an included dimension of
technological accommodation. Intrigued by this challenge and recognizing “a
good mathematical problem still waiting to be resolved”, in 1977, Donald Knuth
of Stanford University decided to tackle the problem of mathematically
designing the “perfect letter”. Systematically approaching this task, Knuth
defined the problems as one of finding the “most pleasing” closed curve, MPC.

He then postulated a set of six axioms to clarify the concept of “most
pleasing”. (Knuth. 1979)


Knuth states the most pleasing
closed curve is one that goes through any n given points in consecutive order,
and the returns to the first point. He postulates six axioms that the most
pleasing curves should satisfy. The first property is invariance, which states
that if any of the given points are rotated or expanded, the most pleasing
curve is to be rotated or expanded in the same way. The property of symmetry says
that any cyclic permutation of the given points does not change the solution,
meaning that if any two of the given points were to switch positions, the curve
would not seize to be a pleasing curve. The property of extensionality explains
that adding a point that is already on the most pleasing curve does not change
the solution. The fourth property is locality, which states that each segment
of the most pleasing curve between any two of the given points depends solely
on those points and the ones immediately preceding and following. According to
the locality property, changes to one part of the pattern won’t affect the
other parts. Taking this into consideration, the search for the most pleasing
curve is simplified, because the problem only needs to be solved in the case of
four given points. This proves to be a great simplification in designing
letters as well, since individual portions of strokes can be dealt with
separately. A way to satisfy all four of these properties is simply to make the
most pleasing curve consist of straight line segments. However, the property of
smoothness implies that there are no sharp corners in the most pleasing curves,
a unique tangent at every point of the curve. Find a way
to describe the math behind defining the perfect curve (with illustrations).


Guided by these postulates,
Donald Knuth derived his MPC as a piecewise continuous curve, where each
segment is determined by an appropriate cubic spline. Appropriate MPCs have
been devised for each letter of the alphabet and loaded into a computer
program. The resulting system, called METAFONT, is flexible and allows its
user, through a variation of parameters, to devise an infinite variety of
letter forms and fonts. Thus the concept of the “perfect letter” now truly lies
within the eye and control of the METAFONT user. Knuth has completed the task
attributed so long ago to Pythagoras, laboured upon by the artists and
calligraphers of the Renaissance and pondered by the Jaugeon Commission. (Show
visual comparison between METAFONT composition and those proposed by
Renaissance masters) Donald Knuth’s resolution of the situation
while ultimately prompted by a mathematical challenge, was also motivated and
facilitated by the existence of new technologies. The scope and power of
METAFONT and its companion TEX typesetting system has revolutionized the field
of typography and demonstrated how mathematics under the rethinking of an old
problem can be applied to new fields. Interestingly, these new resulting techniques
of typography have not narrowed choices as to the “perfect letter”, rather they
have broadened options and personalized the decision of perfection. (Swetz,


A typeface has the ability to
evoke a personality, a characteristic or even a feeling, as perceived by the
observer. This particular point stems directly form perception, thus observers
may not perceive unanimously. Typography has the responsibility of rendering
specific characters representing speech sound and instruction, and “A” will
always be an “A”, but the governing rules and subtle beauties defining the
design of each individual typeface are what influence perception. Erik
Speikermann noted that “if letters are the clothes that words wear, then it
surely follows that there must be as many typefaces as there are voices,
languages, and emotions.” (Mawby, 2016) 


“Like oratory, music, dance,
calligraphy – like anything that lends its grace to language – typography is an
art that can be deliberately misused. It is a craft by which the meanings of a
text (or its absence of meaning) can be clarified, honored and shared, or
knowingly disguised.” (Bringhurst, 2004).

The perception of the voice of typography comes from a richly varied
combination of formal associations with typography, temporal perception in its
reading, and a sense of visual rhythm. (Bringhurst, 2004, p.45)


In 1847, Oliver Byrne wrote and
designed an illustrated book so far ahead of its time that it could have been published
today. The First Six Books of the
Elements of Euclid prefigures the art and design of the 20th
century avant-garde movements. Although the illustration on the title pages is
tremendously similar to that of a de Stijl and Bauhaus design, the mid-19th-century
publishing date disqualifies it from being “modern”. (Heller, 2010)


Every high school student has
suffered though Euclid’s fundamentals of geometry, which is why Byrne stated:

arts and sciences have become so extensive, that to facilitate their
acquirement is of as importance as to extend their boundaries. Illustration, if
it does not shorten the time of study, will at least make it more agreeable.”
(Byrne, 1847)


The colour symbols – circles,
squares, diamonds etc. – effectively substitute for key works, making
comprehension much simpler. By the time the complicated theorems and formulas
appear, toward the end of the book, the reader is fluent in the visual
language. (Heller, 2010)

work has a greater aim than mere illustration; we do not introduce colors for
the purpose of entertainment, or the amuse by
certain combinations of tint and form, but to assist the mind in its
researches after truth, to increase the facilities of instruction and to
diffuse permanent knowledge. .” (Byrne, 1847)