
Lucas Sequences
The Fibonacci Sequence — first recorded in the West during the 13th century by Leonardo Fibonacci — is the following set of integers (whole numbers): 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987… ad infinitum The sequence was named for Fibonacci by the eminent 19th century French mathematician François Édouard Lucas who produced a similar sequence of his own, that he referred to as Lucas Numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843… ad infinitum The Fibonacci Sequence and Lucas Numbers are both specific variations of the generic Lucas Sequence. Lucas Sequences and integer sequences in general have been the subject of a lifetime of study for numerous accomplished mathematicians, so we're going to summarize the concept as follows:- Any given number in the sequence is the sum of the two preceding numbers as described in this formula: n = n-1 + n-2
- The first two numbers in the sequence — known as seed numbers — are selected arbitrarily.
Why use a Lucas Sequence to set type?
Lucas Sequences, particularly in the form of the Fibonacci Sequence bear a close relationship to the Golden Ratio, which is repeated throughout nature in shells, spider's webs, cloud formations and numerous other natural wonders. There is strong evidence to suggest that anything we find visually appealing is due to that form's relationship to the golden ratio. More importantly, Lucas Sequences provide us with a scalable framework for setting type that is both graceful and rhythmic.Determining our first seed number
There is much debate over the correct body text size for a website. 12px is still the most common, but it's not unusual to see anything from 10px to 16px. This is substantially born from a lack of understanding of what font measurements actually mean. Strangely, for those of us accustomed to standardization — or at least aspirations of standardization — there is no standard measure in type design; two different type designers, drawing exactly the same design into font production software are likely to draw it at different sizes. The variation between different typefaces is, unsurprisingly, all too common.
Determining our second seed number
There are numerous strategies open to us for determining our second seed number. Some designers have a specific preference and would pick 1.2em or the like. Other designers, who enjoy the mysticism of the Golden Ratio might like to use 1.618em. A more practical solution is to determine the second seed number using the line-height of the body text. However because line-height is usually dictated by the length of the line and line length is determined by a grid it is more suited to print, or non-responsive web design. Because we're looking to the future, and the future is responsive, it won't work for us. What we're left with, partly by a process of elimination and partly due to its appropriateness, is the type's x-height. Or more specifically, the relationship of the x-height to the rest of the glyph. One characteristic of good design is the repetition of key elements, and carrying the proportions of the type through to the page as a whole is a great opportunity to both acknowledge the work of the type designer and imbue our design with some of the typeface's character.Finding our x-height
To find the x-height of your font, we need to open up something like Photoshop or Illustrator and add some text that includes an ascender (e.g. 'd') and the letter 'x'. If you're using a bitmap editor like Photoshop, set the text as large as possible to ensure you get an accurate result. In these examples I've set the font to 500pt in order to measure it. Next, measure the height from the ascender line to the baseline and the height from the top of the x to the baseline.
Creating our sequence
So, we have our two seed numbers: 1em and 0.716em and we have the formula n = n-1 + n-2: 1 + 0.716 = 1.7160.716 + 1.716 = 2.432
1.716 + 2.432 = 4.148
etc. Which results in the following sequence: 1, 0.716, 1.716, 2.432, 4.148, 6.58, 10.728, 17.308, 28.036… ad infinitum
The design bit
So far, we've picked our way through some basic, but useful mathematics, and we've arrived at a sequence that gives us a range of sizes from 1em to 28.036em and beyond if necessary. We're now able to use these values as our type sizes sequentially to create a well proportioned type scheme:


Flexibility
Frequently, type size is dictated by more than emphasis: issues of accessibility arise if we're talking about an audience with visual difficulties; the sheer volume of content may throw up restrictions; we may need to be flexible — literally — when designing a fluid layout. Fortunately, CSS handles this situation with ease. Because we're using ems for our sizes we can set up our type using our Lucas Sequence and then scale the whole, by setting a default size on the body tag, retaining the rhythm of our design, but increasing or decreasing the actual values.
Finally
As dependent on underlying mathematics as this system is, it's important to acknowledge that design cannot be reduced to a set of equations. This method of sizing type gives you a structure with which to work, and will help you create well proportioned type schemes, but it's down to you as a designer to use the system as a tool, not as a crutch. The notes and scales in music, even Jazz, can be reduced to a set of equations that describe their relationship, but it takes someone like John Coltrane to bring those relationships to life.Ben Moss
Ben Moss has designed and coded work for award-winning startups, and global names including IBM, UBS, and the FBI. When he’s not in front of a screen he’s probably out trail-running.
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