Golden Section as a method for finding regular and bold complements

Nick Job's picture

Thought I'd share this.

I was trying to work out how much thicker a bold weight should be compared to its regular.

I decided to do some research to help, looking at a fairly arbitrary group of 12 font families* (mostly Microsoft) and worked out the ratio between lower case thin and thick downstrokes in each case and then took an average of the 12.

It turns out that the ratios (regular:bold) varied between 0.52 (not enough difference to be effective) and 0.67 (too much difference), but the average was 0.6216.

This turned out to be almost identical to Φ (golden section)


Incidentally, this number is also the ratio between subsequent Fibonacci numbers as the numbers get big.

Since most typeface families have just a single weight in between a regular and its bold, then it follows that the thickness between subsequent weights should increase in increments of √φ = 1.27214 (where φ = Φ + 1) since, using Luc(as) de Groot's method of finding the weight optically midway between two others, m² = b² - r² (where r = regular strokewidth, b = bold strokewidth and m = optical medium strokewidth). Let's suppose r = 1 and b = φ, then m² = φ² - 1², but since φ² - 1² = φ then m = √φ.

It also follows from this that if you have a series of weights 35, 45, 55, 65, 75 and 85 (where 55 is regular and 75 is bold) that the thickness of 75 = the thickness of 35 + the thickness of 55; and the thickness of 85 = the thickness of 45 + the thickness of 65.

You've got to love maths. Any thoughts?

* Myriad, Calibri, Cambria, Constantia, Corbel, Arial, Georgia, Palatino, Trebuchet, Verdana, Helvetica and Ocean Sans

ebensorkin's picture

When I first got to Typophile I was looking at & experimenting with this & I found that while yes, there is definitely something there - like all things in type - it's just one of several factors. No formula can replace the eye. And there are many glyph specific & style specific issues all of which must trump the formula in specific areas. The hyper bold & the hyper thin also impose their own agendas. The hyper thin are difficult because of point placement issues and the hyper bold must be distorted beautifully/grotesquely in ways that the formula doesn't help with at all. AND if you have a font which is screen specific or is meant to be used in a particular media that may trump the value of the formula again - to the point where it may not be useful anymore.

I rememeber the excitement I felt at starting to get ahold of this pattern! Anyway, what I mean to say is I hope this doesn't feel like cold water. I will try to find the thread.

ebensorkin's picture

Here is my old thread. Note the exceptions! Also note jfp's extemely nice examples.

Here are more in no particular order:



If you put this in your Browser where a URL would go you can rapidly serach typophile fot things like 'Making a bold weight' or whatever else you like.


blank's picture

I remember when I thought golden section weirdos were just crazy designers who read some funny old book and lost it. Now I read stuff like this, smile, and slowly take in the ancient workings of the cosmos.

William Berkson's picture

I heard Lucas de Groot give a talk at the TDC in January. His theory is much more elaborate than what he has on the web site. He takes into account the influence of multiple horizontals--eg e,E on what you can do with bolding, and has formulas for the whole thing.

I was impressed with how well he has thought this out, but I didn't take notes. As he gave it out in a public lecture, perhaps he will share his full theory, if you ask him.

david h's picture

> You’ve got to love maths

BTW, if you like this stuff read "Raster Imaging and Digital Typography" (+ a lot of references.)

ebensorkin's picture

Bill, Will you try to bring back some more snippets from your memeory? I am really interested. I have never been able to contact Lucas to ask for more info.

k.l.'s picture

Interpolation theory at
-- click "continue"
-- then click "information" (leftside menu, bottom)
-- then click "interpolation theory" (first section, last link)

Here's a direct link.

William Berkson's picture

Karsten, your link is what I referred to as "on the web site." There is a lot more to the theory, as I said. de Groot regards that page as quite inadequate.

>some more snippets

Well, I only remember the general ideas. Since I am not focused on sans and am not totally convinced about the formulas, I was not concerned that much with the actual formulas.

He started out by saying that if you to the most obvious thing, and add a certain fixed amount to each width to get the next heavier weight, it will look wrong. The problem is that then that if you have a series of say 3 weights, then the ratio of a the lightest to b the middle will be different from the ratio of b to c. To look good, he thinks, you want the ratios of a to b and b to c need to be the same.

That idea is the basis of his formula on the web site. He has generalized that formula to a formula that gives you the relative weights of n interpolated weights, given the extremes.

But he also has a formula for how the horizontals should vary along with the verticals, depending on whether the glyph has 1 (T) 2 (D) 3 (E) or 4 (@) horizontals.

k.l.'s picture

Hello Bill, I must have missed the second sentence in your previous post. As to mere interpolation, many of the more intimidating formulas can be derived from the pieces on his website. I saw a talk he gave recently, and the part about how to treat horizontals is really fascinating! Though I tried to make notes, I wasn't fast enough ...

ebensorkin's picture

I think I am starting to be able to intuitively guess where Lucas is going.... Still, I second Karsten's request. Bill if you could go over that again, that would be great. Feel free to spice it with your own perspective as well! Karsten, can you contact Lucas?

William Berkson's picture

>can be derived

Yes the formula for n interpolated weights can be derived from the formula for three weights. An exercise left to the student :)

The stuff about horizontals introduces new ideas, though, so there's new information there. I also didn't get that so clearly. I probably heard the same talk as Karsten, and it flew by pretty fast, and I wasn't even trying to take notes.

I did get the problem he is addressing with his formulas, though. If you have a lower case c, and fatten the horizontals a certain amount, then try to apply the same to a lower case e, you are going to get into trouble on heavier weights because the counters will start to get too small and clott up. So if you want a rule, then you are going to have to include the number of horizontals as a variable.

As to my own perspective, I am personally concerned more with the question of needed contrast between regular and bold, and in the adjustment of optical sizes, as I haven't yet focused on the multi-weight family.

But to get back to Nick Job's original post, the number of 52% bolder seems to me may be misleading. It might be that this is too much for a bold weight in a serif font that is going to match with text, eg as a side head. And maybe too little for a headline.

For example, Minion semi-bold n stem is 39% wider than the regular. The Minion bold is 62% greater than the regular. With Times the bold is 65% greater than the regular. If you average the weights of the bold and semi-bold of Minion, you get 50.5%, close to your number, but I am concerned that this may not work well for either purpose.

By the way the relation of Minion semi bold and bold, 39/62, also is not a golden ratio, but also not that far off :)

Also I am a bit infected with Bringhurst's view that most bolds are too bold.

I don't claim to know the answers here, I'm just asking.

ebensorkin's picture

Thanks Bill!

enne_son's picture

I have an older version of the Luc(as) de Groot graphic which contains Dutch text that is somewhat different from the English. The Dutch text reads: "Het optisch midden tussen twee gewichten van een lettertype light niet in het mathematisch midden. Een reeks van gewichten wordt opgebouwd volgens de interpolatiecurve."

Translated: "The optical middle between two weights of a letter type doesn't lie at the mathematical middle. A range of weights is built up according to the interpolation corve."

In other words the text on the site misses the preamble."

Tim Ahrens's picture

Yes the formula for n interpolated weights can be derived from the formula for three weights. An exercise left to the student :)

It seems that all the formulae of the Interpolation Theory can be derived from a very simple principle: That of geometric progression.
The Interpolation Theory simply says: "The stem weights are to represent a geometric sequence." Nothing more, nothing less. Sounds less scary and complicated when it's put in words rather than a formula, doesn't it?

However, I see one weakness in treating the stem weight as an absolute value between zero and infinity. I suggest to define "weight" not as stem weight but as black versus white, which means a number between zero (no weight, just white) and one (the point when the counters and white space becomes zero and we have 100% black.)
This would reflect that the geometric series gives nice relations only in the lighter weights whereas in the black regions a slight increase in stem weight can have a significant effect on the "colour" of the font.

William Berkson's picture

Thanks, Tim. That's quite right about the stem widths.

De Groot's formula a = b x b / c can also be written a/b = b/c . In other words, the ratios between successive terms is the same--the defining feature of a geometric progression.

And that's de Groot's basic point in words: don't do successive weights by adding a constant--an arithmetic progression--but by having the ratios between successive weights the same--a geometric progression.

No doubt a geometric progression makes successive weights in a sans *specimen* look nice. But it doesn't answer the question which Nick asked, which is how big a jump between weights is best, nor the general question what weights are needed or best for use as type on the page.

Also the issue of bolding horizontals and curves seems to me a more complex issue, more resistant to simple formulas.

.00's picture

To paraphrase Duke Ellington, "If it looks good, it is good".

William Berkson's picture

>If it looks good, it is good

James, that is certainly true for type, but the question is, looks good in what context, for what purpose?

If I remember rightly, you wrote somewhere before on typophile that your guideline is to do a weight that looks good for text, one slightly bolder that works as text in white on black, and then a semi-bold and bold that work in relation to these two. What it you who wrote that?

I'd be interested in whether you think of the utility of the bolds and semi-bolds that Adobe does, eg, as I mentioned for the case of Minion. Are the bolds actually useful? For what purposes?

For your own Rawlinson, you have four weights. Regular, Medium, Bold and Heavy. Did you design these with specific purposes in mind? Did you follow the principle I mentioned earlier of designing for regular, reversed, etc.?

.00's picture

I did write that, and it applies to serif families I've designed more or less. I think that is what I had in mind when I designed Rawlinson.

As to commenting on Adobe's work, or others, I have to honestly say that I do not look at any contemporary type designer's work. I am not a graphic designer, so I do not use type in that way. I spend many hours a working on my type, the last thing I want to do when I'm finished for the day is look at other peoples type.


Tim Ahrens's picture

Also the issue of bolding horizontals and curves seems to me a more complex issue, more resistant to simple formulas.

I agree. And where would be the benefit of complicated formulas? If we need a special formula for the curves, a special formula for the horizontals of the E and so on there is no point in having formulas at all.

I think mathematics is not a suitable means of defining shapes but it can help us process manually designed shapes, like in MM interpolation, for example.

No doubt a geometric progression makes successive weights in a sans *specimen* look nice. But ...

I totally agree. In fact, sequences with very close "pairs" of similar boldness (like FF Info) can make sense although to a very high degree they do not comply with the Interpolation Theory.

dezcom's picture

“If it looks good, it is good”.


We always have to come back to how the type is intended to be used. If the weights are there to look like a pretty and perfect progression when appearing together on a type specimen sheet, that is one thing. If the weights are there to function as text, reversed text, subhead distinction, and display distinction, it is another thing. All of these wonderful mathematical explanations and methods are terrific for academic discussion and may help us see what could be lurking beneath the core of our vision but, in the end, I don't see how it makes the design process any faster or easier than just using your eyes. There is a difference between the intellectual discussion of type design process and the actual doing of it. This was a wonderful thesis project and fascinating reading but I don't see how I personally would apply it to doing my own work.


Tim Ahrens's picture

I have just uploaded a document that explains the mathematical system for how I generate my intermediate instances:

It is not very elaborate but I hope the point comes across.

I don’t see how it makes the design process any faster or easier than just using your eyes.

The theories do not make anything faster - they need implementation. And theories that cannot be implemented are pointless, I agree.

dezcom's picture

Thanks for the link Tim, I'll take a look at it.


William Berkson's picture

That's interesting Tim, thanks.

This brings to mind the Superpolator developed by the LettError folks.

On Thursday I heard Christian Schwartz talk at the TDC about the new Guardian typefaces that he and Paul Barnes developed. This was a massive project ending up with 200 faces.

When he was asked about how he did the huge number of weights he said he used Robofog, but now that there was a much better program, the Superpolator, and he would certainly use that today. Afterwards I was talking to Hanes Famira who also praised it as extremely accessible and easy to use, and more powerful than multiple masters.

I am thinking I will be able to benefit from some interpolation program pretty soon, and wonder if people have experience using and could compare using the blend tool in Font Lab, multiple masters, and Superpolater?

dezcom's picture

Is Superpolater a PC only program?


William Berkson's picture

It's a Mac only interface, as they explain on the site.

dezcom's picture

Whoops, I didn't see your link. That color just fades into the page.



Nick Job's picture

I understand (the maths at least) from Lucas de Groot's site how an optical "halfway" weight can be established by b² = ac (where a = regular stem, b = medium stem and c = bold stem thickness), although I misquoted it as a² + b² = c² which is Pythagoras' theorem because I wasn't really concentrating!

I also understand that individual letters (such as e) impose crazy complications on the process. I struggle getting bolds and heavies to look right. The heavier it gets the harder it becomes. I also understand that there is a problem with this whole process at the thin and, particularly, the fat ends of a typeface family.

Nevertheless, the real question was "How much thicker a bold weight should be compared to its regular?"

What I noticed from my 'research' (which I know was not thorough) was that as I added more regular/bold examples to the list, the closer the average got to Φ.

James, I completely agree with you and the Duke that “If it looks good, it is good”. But what I found was that when I used Φ, it looked better than the proportions that I was using. I need a lot of help and if maths helps me I'm happy to use it. I want to know how to make it look good/better, I have no problem with your assertion, it's getting there that I don't know how to do!

Incidentally, I realise there was a slight flaw in my explanantion: It turns out that the ratios (regular:bold) varied between 0.52 (not enough difference to be effective) and 0.67 (too much difference), but the average was 0.6216. Of course, the smaller the ratio (regular:bold), the larger the difference between the weights (so the two comments in parentheses above are the wrong way round).

Bill, using your example figures from Minion, if the regular width was 1, then I assume you mean that the semibold is 1.39 and the bold was 1.65. It follows that the ratio I am looking at for regular to bold = 1/1.62 = 0.617 (which is very near Φ). In these terms, the ratio of the semibold to the bold = 1.39/1.62 = 0.858 * (nowhere near Φ and far too large to provide a sufficient contrast to be used as a regular/bold complement together in text).

*The relation of Minion semibold and bold is not 39/62 but 1.39/1.62

William Berkson's picture

>not 39/62 but 1.39/1.62

Yeah, I was comparing the excess over the regular, which is probably meaningless--hence the smiley.

But my question concerned the utility of the Adobe Semi-Bold vs the Bold. To me the Semi-Bold looks better within running text--a side head, a term to be defined, beginning of a list, etc.--and that doesn't have the golden ratio, but significantly less contrast.

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