Proportion: Richard Padovan

Proportion: Richard Padovan — Part 1

I've been trying to get through this book so that I could put up a single post about it but it's 370 pages of detailed information that I think will warrant at least five posts! As I'm almost half way through, I thought I'd mention a few points that Padovan, the writer, has made thus far. Padovan, incidentally, was/is a lecturer at Bath University only a few miles from Bath Arts Uni campus that I attended (so I'm a bit annoyed I didn't find this book then).

The book is, as I say, packed full of really insightful points made by the author and those architects/mathematicians etc he quotes. The following will resemble a bullet point list, unfortunately, until I get my head around it a little more. I can see a mass network of interlinking bits of information coming together from what I read about proportion; architecture; modern researchers like Hambidge, Wittkower and Ghyka; Renaissance architects and painters like Leon Battista Alberti, Vasari, Brunelleschi; the Florentine writers – Ficino, Mirandola, Poliziano; right through to Plato, Plotinus and so on; and then there are all of the lesser related, but significant, subjects like 15th century Poland, the Hungarian Empire, Renaissance Milan and the Academies, Alois Reigl and Nietzsche (whom I also found while writing my dissertation on the Vienna Secession). I began to learn about proportion as an aid to my design work, but I've found more and more than it is intrinsically linked to my other interest – Renaissance Florence/Italy.

Early on in the book Padovan brings up the question of whether proportional systems are to be applied first, as a sort of code of conduct, or last, as a corrective device to improve what has been done up to then. Either way, there are questions to be asked about the process. For instance, can proportional awareness create a monotonous feeling in works? Can it void all sense of talent if used well? What do we mean by talent? Is talent necessary/important? Gustav Fechner proved that the majority of people he chose preferred, out of a large group of rectangles (and a square – being a 1:1 rectangle), the one whose sides followed the Golden Proportions 1:618. Others chose either a rectangle very similar to the Golden Rectangle or the square. What, then, does talent mean with regard to proportion?

Padovan goes on to consider Le Corbusier's/Alberti's idea of the relevance of proportion (in architecture) where the eye cannot see it (in graphic design, this may be a proportional system working very obscurely, in some way). Considering we cannot see, and do not yet understand, all of the universe, we are to assume it is all based on some binding mathematical principal. So their argument is to continue what we assume occurs in the universe by applying an overall principal of proportion to whatever we put into the universe. What is the point of proportion where we can't readily see it? The book says 'the eye of God' but I have a problem with that, if it means the common religious God. Were it to mean the eye of Ficino's idea of the soul, I could agree. In simpler terms, we (I hope) wouldn't feel comfortable tidying a house by taking everything that was out of place and throwing it into a cupboard somewhere. That messy cupboard might rest on our thoughts and give us a guilty feeling. Someone might open it and your reputation as a good housekeeper may collapse! I think this is what is meant in the book by 'the eye of God'. At least, that's how I'd argue for the need to tie every part of a design into its proportional context.

Proportion: Richard Padovan — Part 2

An interesting point made by Hans Van Der Laan: 'The space of nature has three aspects that leave us at a loss…. The sheer fact that we refer to natural space using negative terms like 'immeasurable', 'invisible', and 'boundless' indicates that it lacks something for us. We do not feel altogether in our element within it. Architecture, then, is nothing else but that which must be added to natural space to make it habitable, visible and measurable.'  Something to consider, but I couldn't disagree more with his conclusion. Why do so many people like to 'retreat to the countryside'? I've heard many more people use the word 'beautiful' to describe most parts of nature. If it were beautiful we're saying we need add nothing to it but that which sustains us. Furthermore, it is of nature that we build in the first place. Also, how often do we hear people around the Eiffel Tower saying 'wow', 'incredible', or the French 'incroyable'! (Fun to say). These terms may begin with the prefix 'in' but are not negative – we are praising it and saying how it cannot be believed/measured/understood. And yet, in a previous post, I have already shown the Effiel Tower to be measurable by Phi.

This is why it is so useful to stop ourselves from using interjections like 'wow' and run-of-the-mill adjectives like 'amazing' or 'fantastic' when we see something we like. Next time you see something you 'like', stop and think about it. What caught your eye? The colour? Size? Shape? Message? Idea? Then ask yourself why. It must relate to other things (like objects, or your experiences): what are these things and how does it relate? Answer all these questions honestly and you may find yourself understanding that which, at first, was an allusion to you. I suppose it's similar to how some people fall in love, thinking so highly of one another, only to end up sick of each other a few decades later, after they each know/understand their partner as best they can.

I can't remember why but Van Der Laan's comment made me think about the saying 'it's not what you do but how you do it'. In learning about proportion, it's very easy to start believing this saying, as it relates to Padovan's early comment on whether proportion systems negate talent. If 'how' were all that mattered, 'talent' could be meaningless. Some Renaissance artists have said that the Golden Proportion is not the conclusion of art, but the beginning. It is, according to them, not as simple as creating a pleasing composition. I'm sure the typographer, Jan Tschichold, would agree. Meaning affects composition, and vice versa. If 'how' were all that mattered, does it mean that the work of Dante Alighieri, who was well received because he wrote in Italian with masterful ability (rather that the less practical Latin), loses some of its original merit when translated?

Van Der Laan is again quoted: 'Breathing begins at birth with an inspiration and ends at death with an expiration; so too when we make something we must conceive the influence of forms upon our mind as the initial life-giving movement'. Padovan explains that this is a description of how we operate when building/making. I read it as meaning that whatever we can build or create will be a model of what we have taken in from the world and put together in our mind. This forms one side, anyway, of the narrative that runs through Padovan's book; the tussle between 'empathy' and 'abstraction'. 'Empathy' is the argument that knowing is belonging – what we do we take from nature. What we put into nature is already found there in another form. There are limits that come with the empathetic stance; we are only as advanced as nature allows us to be. 'Abstraction' is the opposite extreme whereby human beings have the ability to operate independently of nature and our architecture etc is imposed upon it; for example, the square, accord to supporters of 'abstraction' is not found in nature. I think there may be another question to it; does nature not include the human mind? Where does nature end and living organisms begin? Surely the human mind and its capabilities are part of nature. A square, then, is conceivable, and part of nature. After all, nature does not end with what we see. 

I recently read something written by Marsilio Ficino, a Florentine during the Renaissance, that may add to this: 'Do you desire to look on the face of good [not God]? Then look around at the whole universe, full of the light of the sun. Look at the light in the material world, full of all forms in constant movement; take away the matter, leave the rest. You have the soul, an incorporeal light that takes all shapes and is full of change. Once again, take from this the changeability, and now you have reached the intelligence of the angels, the incorporeal light, taking all shapes but unchanging. Take away from this that diversity by which any form differs from the light, and which is infused into the light from elsewhere, and then the essence of the light and of each form is the same; the light gives form to itself and through its own forms gives forms to everything.'

A further point that Pandovan touches on, and the final one I'll mention here, is the need to combine unity and complexity to create 'true order', where 'unity' is a constant and 'complexity' is an unexpected outcome. For instance, to work with irrational numbers like 1.618 is to give 'complexity' to the 'unity' of the number 1. Combining a rational number like 2 with the 'unity' in 1 is not a method of true order. Jay Hambidge puts the two outcomes down to 'static' and 'dynamic' symmetry; irrational numbers will create unexpected outcomes in proportion systems, like Phi, and are therefore the creators of a 'dynamic' symmetry. Rudolph Wittkower, similarly, puts the two distinct methods down as the 'geometrical' and 'arithmetical'. This idea of unity with complexity can be found in other formats though; even just to look at the decorative elements in Renaissance armour, as shown in a previous post, we find 'unity' in the fact that the illustration sits upon a central mirror axis and 'complexity' in the actual content of the illustration.

I did read about Van Der Laan's recording of the 'plastic number' and have found that I came across it myself by accident in a previous post. I didn't know it was so significant at the time but I'll have to repeat the post and add into it what I've read in Padovan's book. So, more to come on this, for sure.

Eiffel Phi

Hmm. Thought so…

Phi in the face of beauty

The discussion on the relationship between Phi and the concept of beauty is an old one – and I doubt we would be brave enough to make any conclusion about it in the near future. It’s a somewhat worrying concept to many people: the idea that a number may be behind all that is considered beautiful; that beauty is not subjective, not a case of each to their own. Because Phi is arguably rooted in all nature, it could be that we decide something is beautiful because we have seen its proportions a million times already – and we like familiarity.

I have seen examples of Phi applied to the human face before and it is apparently a champion female tennis player whose face was most commonly used. I’m interested in the role of Phi in beauty though – and if I think of the most popularly beautiful person in recent history, I think of Marilyn Monroe.

I should explain as usual, for any readers unfamiliar with the subject, that Phi is simply a number (1.618) to be used to make proportions by the ratio 1:1.618. It’s also known as the Golden Section or Golden Ratio.

The images below speak for themselves. Look for yourself and all of the ways the lines complement the face. The significance here is that one number, Phi, is behind the relationship of every single pair of lines drawn. By mostly using the measurement between Marilyn’s chin and hairline I have drawn out the golden rectangle and divided it by the proportion 1:1.618. All of the lines below are proportional by that ratio and yet they play to Marilyn’s features incredibly accurately. Even the curve of her hair follows the pink Golden Curve when it spirals from her eye. She must have had a good stylist.

The penultimate image below tries squares as its marker. I began with one whose centre would be Marilyn’s eye and whose edge would touch the base of her chin. Then, reducing in size by 1.618, the squares even show us that her beauty mark is not out of place. We could try an endless number of shapes here to illustrate the wonder of Phi but they’re all fundamentally the same thing – lines whose positions relate b Phi.

The face is not the only human feature to have been analysed this way. In fact, Renaissance artists who dissected bodies (not only Leonardo) would have made such research using Luca Pacioli’s findings. Have a look at your hand – do you see how the ratio between the length of the palm and the length of the fingers looks roughly 1:1.618? Phi is all over us!

Ubisoft logo

I noticed another example of the use of Phi in design – the Ubisoft logo. The circles are diminished in size by 1.618 (Phi). I wonder if they wanted it to relate to their business or just look good…it’s pretty much a golden spiral.

New numbers

In my last post I pointed out how the ratio 1:1.618, manifests itself in segments of a circle – of any angle. Three segments of a circle – of the relative sizes 1:1.618:2.618 – can be placed in such a way that the two smaller segments will sit inside the larger to create a curve with their edges that will touch the straight edge of the largest segment. I thought it would only be 1.618 that could create such an effect but I found that the numbers below do the same thing.

The first number I found was 1.465. The Fibonacci sequence, which creates numbers relative by 1.618, relies on your adding the two previous values to get the next (in the sequence 3, 5, 8, we get 8 by adding 5 and 3). If you keep calculating the Fibonacci sequence into the hundreds of thousands, you‘ll find that the relation between the numbers gets closer and closer to the complete number of Phi – though only the first few decimals really matter. 8/5 is only 1.6, so it’s necessary to keep going through the sequence until it gets a more accurate value for the relation between numbers that are the sum of their two precedents. If 1.618 is gained by adding the first and second values in a sequence, 1.465 is made by adding the first and third. If you started with 1, 2, 3, then, the next value would be 4 (3+1). If you continue the sequence you eventually get the numbers 872, 1278, 1873 and 2745 (2745=1873+872) and if you divide 2745 by 1873 you get 1.4655632…. You could round it to 1.466 or just leave it at 1.465 – you’d have to keep going in the sequence to check if it was going up or going down as it averaged out.

Anyway, the first image shows what 1:1.465 does with circle segments relative by its value – remember the second smaller version is ditched – its the first and third smaller versions of the largest that we use.

Adding the first and fourth previous numbers in a sequence will give you proportions of 1:1.382 and if you use the first and fifth previous numbers, you get 1:1.324. Interestingly, in the 1:1.324 sequence, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 106, you’ll find that, as expected, 34 = 26 (first previous) + 8 (fifth previous) but if you add neighbouring numbers, as you would in the Fibonacci sequence, you get a similar effect to the Golden String – 3+4=7, one over 6, 4+5=9, one over 8, 5+6=11, equal or 0, 6+8=14, one under 15, 8+11=19, one under 20, 11+15=26, equal or 0, 15+20=35, one over 34…and so on. In other words it goes +1, +1, 0, -1 -1, 0, +1, +1, 0, -1, -1 etc.

There must be some significance with these numbers if they act similarly to Phi with regard to circle segments within related segments…but I’ve yet to find it out. I added 1:1.414 (the useful proportions of DIN paper edges) as well, to demonstrate how it conforms with the seemingly special ability of these numbers to make segments of any angle able to relate the way they do in the images here.

Golden spirals

Some quite heavy-on-the-eye geometry made to look pretty here. I got a little confused when I read about the Golden Spiral – it made me think there was one, and only one, Golden Spiral but, in fact, it is the construction which decides if something is a Golden Spiral or not – and not the angle it’s drawn with. Usually a Golden Spiral drawn with a 90˚ angle is said to be the Golden Spiral – the one you see inscribed within a Golden Rectangle. A Golden Spiral can actually be any spiral drawn using similar segments of a circle which are reduced by Phi and arranged as shown in the first image below.

The blue triangle is similar to the others, but smaller than the magenta triangle by the ratio 1:1.618. You can guess how the green relates. The second grouping in the first image here shows the important characteristic of Phi – it is the only number capable of creating a sequence of numbers that relate to one another by addition and multiplication – in 3, 5, 8, 13, 5+3=8 and 5x1.618=8 as 8x1.618=13.

Phi can be found in a sequence in which any number is the sum of the 2 preceding ones. I tried creating a sequence of numbers whereby any number is the sum of the 3 preceding ones by choosing any three numbers to begin with and working through the numbers. It doesn’t matter what numbers you begin with because the more you add numbers to the sequence, the more stable the ratio will become and the closer you’ll get to finding the special number. I got 1:1.839 as the ratio for a sequence relating to every 3 numbers. 1:1.926 was reached by looking at a sequence using the preceding 4 numbers. If you look at the second and third image below, the groups to the right show jagged stars marking the failings of these ratios to act in a similar way to Phi.

The fourth and fifth images show Golden Spirals in steps of 10˚ up to 360˚. Some significant spirals occur at 36˚, 60˚, 72˚, 90˚, 108˚, 144˚ and 180˚ – 60˚ giving a hexagon, and 72˚, a pentagon, for instance. It’s interesting how wonky they get after 180˚ – perhaps it’s not necessary to go beyond that point though, because you can no longer join up the points of the curves with straight lines.

The sixth and last images demonstrate the pattern of end points in golden spirals – all ending on the circumferences of circles (in pink/dark pink) which are reduced by Phi from the largest. This is expected, really, but in the sixth image, where I’ve reflected all of the spirals up to 180˚, we find them producing proportionally smaller circles by their interactions. All this can be anticipated with an understanding before hand, but seeing it at work is something else.

Pentagram construction inside golden triangle

Another quite slick construction here.

Expand a line by the ratio 1:1.618 (using the golden rectangle from a square method) and then extend arcs as shown in the first image. You can then produce a golden triangle by joining the corners – the magenta arc makes sure two sides are of equal length, and the blue arc makes sure it’s a 72˚ angle – meaning the third edge will be to the others as 1 is to 1.618.

Find the mid-points of each edge to the triangle. By joining these points we have the all-important triangle with which to create a pentagon/pentagram.

As we know the sides are going to be of equal length we can just use the compass, like in the fourth image, to find the two remaining points to the desired shape.

I added a last image to show the relation of one side of the pentagon to the longer side of the triangle – if the triangle edge is 1 unit, a side of the pentagon is 0.5/1.618.

Nothing accidental: Edvard Munch

The Scream. A 20˚ golden spiral. Simples…

Another golden pentagram construction

Here’s a nifty little way to draw up a set of golden pentagrams – which are, between themselves, proportional by 1:1.618. I’ve not seen this method elsewhere…and there’ll be loads of ways to construct these shapes but I find this one quite enjoyable to work through.

Where other methods I’ve found (have a look in the February posts) have begun with the golden rectangle, this one begins with a square. The key to drawing the five-points of the pentagram (five-point star) on a circle’s circumference is obviously in figuring out the split of the circumference by 5 (5 equidistant points after all). So we need to be able to chop a circle into 5 equal angles without using measuring tools.

Beginning with a square, we draw a circle whose centre is at the corner with a radius equal to the square’s length. Then we extend the square into the circle by making it a golden rectangle, as described in the second image below.

In the third image we have our special measurement (the dashed line) – one that will cut our circle into 5 equal parts. It happens to be the diagonal measurement inside the segment by which we extended the square. Continue the line, in a circle, to the edge of the main circle. At this point it doesn’t matter where we put our compass – you’ll notice the end result is a pentagram on its side so if you want one with a horizontal base, place the compass at the top of the circle’s circumference in the centre, instead of at the square’s corner.

We’ll make two new points on the circumference by continuing the line right round. These points mark where we’ll put the compass to make further guides. The fourth and fifth images show this process – and we get a beautiful shape at the end of it…a sort of bloated group of diminishing pentagrams and pentagons.

In the last image, the points are simply drawn together with straight edges. The conclusion is an image that harvests squares, triangles, circles, pentagrams and pentagons all in a number of sizes in proportion of the ratio 1:1.618. Further points can be joined, more lines extended and even more interesting patterns uncovered.

Book: ‘The Divine Proportion: A Study in Mathematical Beauty’ by H.E. Huntley

I got this book the other day – along with a few other £5 gems on Amazon – and found it a really good buy. I posted Ghyka’s book, The Geometry of Art and Life, before and, although it was a good collection of examples of where the golden section comes into play, I still wanted to actually learn more about Phi.

This book was ideal for getting a better understanding of the numbers themselves and how everything works together. It was written in the ‘70s and there is quite an amusing passage where the author discusses a computer, which he needed permission to use, that could give him the answer to an enormous calculation he was making. If only he knew how much easier it would become in a decade or two. Still, one of the points he makes in the book is that there is still much to be uncovered about the ‘beauty of maths’ and, with the constant evolution of technology, this is no doubt true today.

The book covers the golden section/divine proportion/Phi in a vast array of drawings and calculations. Some of the mathematics is a little advanced for a modest GCSE-level pleb like myself, and Huntley happily points that out in his book, but with his equations in front of you, it isn’t difficult to learn the necessary parts of algebra involved.

I love the way Huntley writes the book – it’s seriously in-depth and yet light-hearted and personal. He makes a great attempt at rationalizing the idea of beauty and why Phi is the most pleasing formula to the human eye. Is it that you learn to love home, and home happens to be the world in which Phi plays so big a role? Or is it subjective?

Huntley makes the point that beauty may indeed be in the eye of the beholder but there are two levels to our perception of beauty. One is inborn in us all – that sense of ‘judging the book by its cover’. Then there is ‘acquired’ understanding of beauty which means that we find beauty through education – I can vouch for this in some ways: I remember an art history lesson involving a painting of Napoleon. I saw the quite plain-looking painting and got quite bored with it until my teacher pointed out the humour behind the image – the artist had an agenda – propoganda – and had therefore added a quite large shadow to the figure’s crotch, basically indicating he was well-hung as if to say ‘he’s the daddy’! Being a dippy teenager, I found this hilarious and have remembered that piece of information ever since. So, because I know that little fact, I feel more attached to the painting – even more so than, say, the Eiffel Tower because even though the latter is more immediately beautiful, I have the ‘acquired knowledge’ in Napoleon’s portrait and it works as added beauty.

Huntley talks about poetry and music and relates the two to the golden section – apparently in music there are time intervals which interact with the body’s internal clock in such a way that we find them the most pleasing of all. This interval happens to be in direct relation to Phi, telling us that the golden section has a claim to us humans more intrinsic than that which was illustrated by Renaissance artists with the human body, namely a connection to the soul itself.

Huntley quotes Francis Thompson to make a point for those who put little value in the matters that he discusses in the book (for lack of education or knowledge, as he argues):

The angels keep their ancient places;
Turn but a stone and start a wing!
‘Tis ye, ‘tis your estrangéd faces,
That miss the many-splendoured thing.

Max Bill’s circle

I tried using phi to make the yellow circle, instead of the 0.5 that Bill used, and found that the first version (second image) is a little less well-balanced in comparison with the original. The second version, however, (last image) I think is pretty successful. The smaller the circle, the more the text box comes up – which is what you’d want on a poster. The image would still be seen first from a distance.

I’m assuming Bill was making a geometrical representation of a Zulu shield with his image (the words ‘negerkunst’ and ‘sudafrikas’ are a clue straight away). The smaller version of the circle, then, would perhaps make more sense. It gives the image more space and is clearly smaller than the text box – avoiding any conflict between the two shapes.

No idea why the colours are faded on this screen but if you click on the image, it’s better.

Nothing accidental: Graphic Design

I worried that all of my geometrical ‘constructions’ may be misinterpreted so the following examples of geometry used in design may put things into perspective. The point of making those constructions, like the golden circles, pentagrams and so on, is to find methods of drawing proportionally-related shapes without measuring anything (cheating?). Doing that will make it easier to understand how proportion works and also allows you to work out interesting layouts or even grids.

I have, in fact, stolen the examples below from a book about geometry in design (by K. Elam, which I’ve posted about before). They may be a little aged but the two were done 50 years apart and geometric proportion will never be irrelevant to graphic design. The posters are ‘Folies-Bergère’ by Jules Chéret and ‘Negerkunst’ by Max Bill, respectively. I think I’ve added to Elam’s analysis of Bill’s work here because she credits much of it on root 2 proportions – I found that the golden section works on it quite strongly.

In ‘Folies-Bergère’, then, we can see from an immediate use of rectangles which are smaller than the frame by phi (golden section=1.618) that the three characters are divided by these measurements. The second image shows a pentagram (five-point star) within a pentagon within a circle. The whole object is central in the frame. The edges of the pentagon are of a measurement which is equal to the frame breadth/1.618 (phi). In other words, the artist, presumably, used a pentagon whose sides relate to the frame breadth by the golden section. The pentagon works as the master guide – marking where the text will sit (above and below), and informing the angle of the dancer’s legs and the position of the elbow.

Bill ’s poster is a little more complicated but the images I have made here should speak for themselves. If you divide the non-golden rectangular frame by 1.618 you get rectangle A. It isn’t always necessary to work inside an area relative to the golden section but probably an advantage. I’ve wondered why Bill chose to position the shape where it is and can only suggest it is the mid-point of rectangle A when turned on its side. Turning it this way will not affect the proportions in use.

Divide rectangle A by 1.618 and you get the measurement used by Bill to draw his text box. The text box is actually square but this is still proportional to the rest because a square is a phi rectangle with a phi² rectangle next to it. (Phi² is 2.618, as opposed to 1.618 and therefore makes for a slimmer rectangle). Squares mean ‘1’ and are therefore neutral in terms of proportion. The text box is aligned by its left side to the centre of the circle above and is therefore a little cushioned from the edge. Bill may have chosen to move it up from the lower edge also – to give it some ‘breathing space’.

The size of the circles was a little unclear to me at first. The shape in the poster is basically two big circles placed in such a way that a circle of half their size fits into the area where they overlap. The large circle is, as shown, the difference between the corners of rectangles A and B. The smaller circle, as mentioned, is half the size. Why Bill chose to make the smaller circle half the size is a slight mystery. Halfs, in proportion, are known as ’static’ proportions because they fail to give aesthetically pleasing results. Phi, meanwhile, is a ’dynamic’ number because it gives proportions which are both systematic and beautiful.

The reason Bill chose to use a static proportion for the small circle may be because the exhibition was about art which used these proportions (and his geometric shape is a representation) or because using phi may make the shape conflict with the text box. I may re-design it by using only dynamic proportions, just to see what happens….

Proportion in typography

When I was writing the Seurat post I mentioned that, in that incidence, the difference between one number, 1.618 (the golden section) and another, 1.414 (DIN paper system), didn’t make much difference. In other examples, though, it will, and below shows how.

I’ve used the Fibonacci sequence for a long while as a guide to my type sizes because, when I was just starting out in using type in my designs at uni, I read it would give me the most pleasing proportions. It has stuck with me since but I now know much more about it. The sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 etc, is made by adding the previous number onto the current number (starting with 1 and adding 0 to get a second 1). If you take a number in the sequence and divide it by the preceding one you’ll get something close to 1.618 (golden section). Notably, 8 and 13 are in the sequence (13:8=golden section ratio). The further you go in the sequence, the closer you get to 1.618 in dividing each number by the one before. In other words, the Fibonacci sequence is pretty much the golden section (it’s a rounded up/down version).

Not only that, you can start with any number, like 10, and add, say, 5 and repeat the process, (so 5, 10, 15, 25, 40, 65, 105, 170, 275, 445, 720, 1165) and, if we divide 1165 by 720 we get 1.61805555…. Golden section. Spooky! What’s strange is that you’re not using a fixed number to enlarge the previous – 10 is 2x5 but 445 is 1.618181818x275.

The two sets of lines below do use a fixed number to increase the type sizes. One is the golden section (right side) and the other uses 1.414 (the ratio of DIN paper measurements). Although the golden section sequence looks better hands down I wonder if it would still be most suitable to use the 1.414 sequence if you were working on A4/A3 (DIN) paper. Even lettering would be a consideration – lots of lettering is based on the square, but would it not be more logical to design type whose character shapes were similar to the shape of A4 paper?

Five-point star: second method

In the post called Golden Pentagon Construction I make a five-point star with the method most often demonstrated by writers on the subject. The method below is another I have drawn up myself and is a little more direct than the aforementioned – though the shape will be the same size, either way it is drawn.

The first image shows the start of the construction. I begin with a golden rectangle and divide it on one side into smaller golden rectangles. One length in the five-point star is the right hand vertical edge of the largest rectangle. We know, then, how long each line in the shape will be.

The difficult part of making a pentagram is figuring out the angle of the points. The angle, from the top right corner, happens to be one which will lead the line to the corner of the smallest rectangle in the image – it is reduced by 1.618 four times from the largest one. How long you draw the line toward that corner is already clear – the length of the largest rectangle. However, it happens to be a the horizontal halfway point of the second largest rectangular section.

The process is repeated, as in the second image, for the upper half of the shape to get the black lines in the third image. Then, taking ignoring most of our divisions of the largest rectangle, we divide the right vertical edge by the golden section (the square inside the golden rectangle does this immediately). This 8:13 (golden ratio) point on the right edge is our intersection point for the other lines in the pentagram – shown in light grey. The lines can simply be extended until they meet or drawn until they match the length of the vertical edge of the rectangle.

The points are joined very easily, as in the last image, and the outer lines of the pentagon can be formed as a result. This is perhaps even easier a construction than the that of the other post but lacks the advantage of creating an object which is in parallel with the original rectangle.