Credit cards are in the shape of a Golden Rectangle. If you ever need an easily accessible example of a golden rectangle illustrating the proportions of the golden section, all you need do is to pull out a credit card or drivers license.
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Standard sized credit cards are 54mm by 86mm, creating a ratio of 0.628, less than a millimeter off from a perfect golden ratio or golden section of 0.618, the reciprocal of 1.618.
Your credit history
Credit cards were created in 1951, with the first plastic version coming in 1959. The early credit cards didn’t have golden rectangle proportions. That came sometime later. The earliest ISO reference I can find is from 1980, but there likely was other use of the golden rectangle in credit cards before then.
https://time.com/4512375/first-credit-card/
https://bankerandtradesman.com/this-month-in-history-the-first-credit-card/
I was looking for the standard size of Credit card. Are you sure the size you have mentioned is correct!!
Yes. See https://en.wikipedia.org/wiki/ISO/IEC_7810. The official US size is 53.98 x 85.60 mm, so the 54×86 stated is correct to within fractions of a millimeter. It’s not exactly a golden ratio, but close enough for making quick very close approximations.
The Golden Mean or Ratio appears in mathematics and in nature. https://en.wikipedia.org/wiki/Golden_ratio
See also our articles on mathematics, life and beauty.
The above-stated credit-card dimensions differ from the golden ratio by about 2%..
If the golden ratio were intended, then there wouldn’t be that 2% departure from the golden ratio.
But the close similarity suggests that maybe that shape is perceived as neither too long nor too short
Michael Ossipoff
As an update, after this article was first written, ISO standards were released in 2003 that state the dimensions of a credit card to be 85.60 mm x 53.98 mm. These numbers were apparently converted from the English sizes 3 3/8″ x 2 1/8″. The golden ratio of 85.60 is 52.90. So while that is a 2% difference, it’s only a difference of only 1 mm. This is not visibly perceptible, so the credit card still serves as a nice, handy reference for evaluating golden ratio measurements.
It’s a near perfect Golden Rectangle
86mm : 53,75mm
I don not think Phi determines the ratio of a true golden rectangle, that’s reverse thinking.
21/13, for instance, is a perfect golden rectangle; not 21,034/13, which is closer to Phi. (1,618)
You appear to be confusing a Fibonacci spiral with a Golden Spiral. The Fibonacci spiral has rectangles and rectangles based on Fibonacci numbers, such as 21 and 13. This only approximates a Golden Spiral, which is comprised of true Golden Rectangles. A Golden Rectangle has sides in the ratio of Phi (1.618…) to 1.
According to the definition of a Golden Rectangle you are right. (A Golden Rectangle is a rectangle whose width is to its length as the length is to the sum of the width and length).
Unfortunately such a rectangle is just an ideal; it is completely irrational. ?
Being irrational really has no impact on our ability to apply the concept. A 1″ square is “just an ideal” too, because in practice it will never be EXACTLY 1.000…” with an infinite number of zeros.
That’s just the nature of the physical world in which we live, and there’s nothing unfortunate about it. We can apply mathematical concepts with as much accuracy as needed, and that’s rarely beyond about 5 decimal places.
I assume we agree, that the credit card can be considered golden even thought it does not exactly match the (in)definite golden standard: 1,618..x 2,618.
I tend towards a more discrete definition of the golden rectangle as rabbits do not produce fractions.
We have to be careful to not call anything that comes close to 1.618 to be an application of the golden ratio. Conversely we shouldn’t say that an informed intentional application of the golden ratio, executed with reasonable accuracy, is not a golden ratio because it wasn’t done with infinite precision.
The problem with the “discrete” application based on integers and rabbits is that it is highly inaccurate early in the series. The ratios of the first successive Fibonacci number pairs are 1, 2, 1.5, 1.6 and 1.625. It’s not until you get to Fibonacci numbers of 21 and higher that you get ratios that I would consider close enough to be considered “golden.” The first Fibonacci sequence pair to produce 1.618… is 233/144. That’s a LOT of rabbits!!!
I just don’t think that I am the one who is ‘confusing’ the Golden Rectangle with a Fibonacci one. Seems to me they are one and the same. I understand that the ratio is primarely quite ‘far’ of from being golden, but are it not the Fibonacci numbers themselves that have the unique interwoven property that generates the golden ratio.. Even if you take 2 Lucas numbers for instance; they still grow in the same ratio:
3+4
3+4+4
3+3+4+4+4
Or any number
Anyway, thank you for your brilliant site.
A Fibonacci spiral and Golden spiral are similar, but uniquely different, as shown here: https://www.goldennumber.net/spirals/
The Golden rectangle of the Golden spiral is always the same in its proportions. There is but one!
The Fibonacci rectangles of the Fibonacci spiral though change in proportion with every new set of Fibonacci numbers, and there are an infinite number of them. So if we refer to a “Fibonacci rectangle” which one do we mean? The Fibonacci rectangles get closer and closer to the Golden rectangle as the series progresses, but it’s never the same as the Golden rectangle.
I’ve not heard though of an
It is rather doubtfull to state the ratio changes while the the initial difference of 1 is ‘nihilated’. The definition of the ratio’s becomming one is not an exclusive one as this also acoounts for all 1/x + n recursions.
A Fibonacci spiral and Golden spiral are similar, but uniquely different.
I do not think such a sharp distinction between a perfect golden rectangle and lesser ones can be made. One can always add another square to form a more perfect golden rectangle (/ratio).
Very true. I cover this at https://www.goldennumber.net/spirals/
ps. no more dan 13 rabbits are needed to come closer to phi than with a creditcard.
5/8 – 0,618033 = 0,006967
53,98/85,60 – 0,618033 = 0,012574
The comment-box is 5,60 x 2,85 on my device; that’s too far off ?
A bonus feature is that 5+3 and 8+5 are sums of Fibonacci-numbers.. Using inches (3 3⁄8 × 2 1⁄8), the ratio is 34/54. Only one count off
A dimension of 53,4 : 86,4 would have been even more perfect, as it is equal to 8,9/14,4 = 0,618055555…
Very true, but then I don’t think they were interested in trying to express the dimensions of a credit card to 1/10th of a millimeter.
The explanation, that credit card’s dimensions reflect the gooden ratio is not plausible.
The ideal approximation for the golden ratio is to divide two adjacent Fibonacci numbers by each other, the larger by the smaller.
The numbers 4181 and 2614 – fibonacci – yield a better approximation (1.599) than 8560 and 5398 – credit cards)(1.586).
Admitted, the resulting credit card would prove rather small. For a more convenient size, one coul preserve the ratio by multiplying both fibonaccis by two. Giving a length of 8362 and width of 5228 which is quite close to the current size.
Using imperial measurements, you can find similar fibonacci approximations.
In my opinion, the credit card dimensions are arbitrary. The possibility remains, that the original designer approximated Phi by using a fibonacci divider compass, a common artist tool. The result was of course mathematically unfounded and therefore inaccurate.
The article does not make any representation that the design of standard credit card dimensions was based on the golden ratio. If that were the case it of course would have a length to width ratio of 1.618 rather about 1.59.
The point of the article is simply that a credit card gives a good, quick approximation of the golden rectangle with its sides in a ratio of 1.618… to 1.
Given that, there’s not reason to seek Fibonacci numbers as these just converge on the golden ratio anyway. In your example, 4181 is a Fibonacci number, but 2614 is not so that doesn’t help much.
If we accept 85.60 as the industry standard width, the height at its approximated golden ratio would be 52.90, for a ratio of 1.61814745…, which is visually imperceptible from 1.6180339887…
That means the credit card is about 1 mm wider than a true golden ratio, which again makes it a good, quick guide for approximation.
One might just as well question the arbitrary ‘fact’ of whatever dimension has been given to creditcards & the like. One determinator was probably if it would fit into a wallet and another more ergonomic if it was handy, fitting nicely within the palm of ones hand, like Jain108 demonstrates in a youtube video. Myself I have come to assume the ratio is a mix of the golden ánd silver ratio, connecting it to the A-standard paper, although I am not sure what came first. Anyway, 2- 8,560/53,98= 0,4142..
Acknowledging the fact that the credit card is quite a good approximation of the golden ratio, the question remains, if there is any significance to it, or if it is merely a random fact or not, as the ratio is regarded by some or even many as a physical expression of some (semi)magical force, ordering the universe. There is even someone who claims one can draw a scientifically sound life chart on the basis of it; which to me seems a quite far-fetched.
Another (little) curiosity is, that if one cuts off a 53,98 x 53,98 square from the creditcard and repeat the proces with 31,62 (85,60-53,98), one will end up (down) with a 31,62 by 22,36. An approximation of the root of 2, with both sides being the first 4 digits of √5 and √10 x 10. That seems quite significant to me.
2,236 (√5)
3,162 (√10)
5,398
8,560
Thus, I do not believe the likeness popped up without intent, as that would be statistically absurd and mean that even a (1) monkey could have chosen the creditcard to be what it is; standard paper (DIN476) with a Fibonacci-twist.
As a consequence; 4 sheets of A4 paper could be used to imitate the creditcardshape: (2*29,7×21) : (29,7x21cm) = 80,4: : 50,7
I meant 6 sheets of A4, with a gap in the middle that magically holds 7 creditcards. since 29,7-21 = 8,7 and (2x 29,70 – 21) /7 = 5,48
Here’s someone blaming Donald Duck (i.e. Walt Disney) for perceiving the credit card as a Golden Rectangle.
Quite strange, as the main theme of his books and talks, is that accuracy is rarely needed to find a fairly correct answer.
https://robeastaway.com/blog/golden-rectangle
ps all the more weird is the other visitor’s (personal) preference for ‘a’ ratio of 1: 1,41 (due its property of staying the same when doubled or halfed) is acclaimed to be as far from (mathematically) preferable to let’s say anyone as the Golden Ratio, which is pure nonsense.
The unlikely “closet mathematician” referred to in the article, might just have been an equally big fan of the silver ratio as well with a rational credit card ratio of 157/99 not only being close to Phi, but also a silver rectangle, i.e. 41/99, from a double square.
In general I find it strange to suggest that 27/17 is no way near the golden ratio being only 1/17 off from the fairly accurate approximation of 55/34
Another obscure fact is the creditcard’s relation to the great pyramid of Gizeh. A monument that appears to be based on 2 Fibonacci-rectangle of 89:55 ( i.e. join the 2 of them together so you get a 89:110 and drop both heights to the middle at a common height of about 70.)
If you use 2 creditcards instead; you won’t get a Kepler pyramid, of course, with a base to height ratio of 7/11, but a Creditcard Pyramid with a base to height ratio of (exactly) 8/13 !!
Was the designer of the creditcard maybe a Freemason and or a huge fan of the great pyramid of Gizeh ? I wonder.