Golden Ratio, Phi and Fibonacci Postage Stamps around the World

Governments around the world commemorate the Golden Ratio, Phi and Fibonacci sequence.

Stamp collecting is one of the world’s most popular educational hobbies, and is enjoyed by people of all ages and nationalities. The most popular stamp ever is US 1993 Elvis stamp, with over 124 million collected. So, the Golden Ratio and Fibonacci sequence may not compete with Elvis in popularity anytime soon, but they too have been recognized with commemorative stamps issued by governments around the world. In addition to adding some “phi” to “philately,” they offer some interesting expressions of golden ratio concepts, as well as its application in design.

Liechtenstein 2013 Commemorative Fibonacci Sequence and Phi Stamp set:

The Principality of Liechtenstein, a landlocked micro-state bordered by Switzerland and Austria, issued a set of three stamps in 2013 that illustrate the Fibonacci sequence and its relationship to the golden ratio.

Note that interesting presentation of concepts:

The first stamp lists the numbers of the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, etc. Alternating bold letters were used to show the numbers.
The second stamp shows the ratios of each successive pair: 2:1, 3:2, 5:3, 8:5, 13:8, 21:13, etc. Alternating bold letters were used to show the pairs.
The third stamp shows the golden ratio (1.6180339887… ). The ratios of the Fibonacci pairs converge on as the series increases. Curiously, the decimal representation of Phi (Φ) on this stamp has two errors. 3653 should be 36563, and 0911 should be 0917. These differences would not impact anyone’s practical use of the number, but here they are:
- 1.61803398874989484820458683436563811772030917 is the golden ratio, as rounded.
- 1.6180339887498948482045868343653811772030911 is shown on the stamp.
Note that the final numbers in purple, green and red are denominations and not part of the Fibonacci sequence or digits of Phi.

Note: Click on the images on this page to see the full resolution images.

Use of the golden ratio in the design of the Liechtenstein stamps

Interestingly, the designer very creatively applied the golden ratio in the design of the stamps as well. This is shown in the grid overlays created with PhiMatrix software.

Stamps 1, 2 and 3 (top row): The center stem of the leaf in all three stamps is positioned at the golden ratio of the width of the stamp. The point where the stem meets the leaf is also at a golden ratio of the height to the stamp.
Stamp 4 (bottom row left): The purple leaf design within the circle aligns to golden ratios from the center of the leaf to the outside of the circle.
Stamp 5 (bottom row middle): The points of the purple leaf aligns to a template based on the golden ratio. The concentric circles, divided into 36 sections, have diameters are based on the square root of the golden ratio. This means that every other circle is a golden ratio of the diameter of the one before it.
Stamp 6 (bottom row right): The dimensions of the stamp itself are a golden rectangle. The blue rectangle inside perforations has a length to width ratio of 1.618.

All in all, a rather very creative and thoughtful expression of the golden ratio and its application in design.

Macao China 2007 Commemorative Golden Proportion Stamp set

Macao China in 2007 issued a two sheet series called “Ciência E Tecnologia- A Proporção Dourada,” or “Science and Technology – The Golden Proportion.” Macau, spelled Macao in English, is officially the Macao Special Administrative Region of the People’s Republic of China. It’s an autonomous territory in East Asia, bordered by the city of Zhuhai in China and the Pearl River Estuary.

One sheet has just a single stamp, surrounded by illustrations. It has the digits of phi as its background. The other sheet has 8 stamps. It has a nautilus shell as its background.

The full sheet has four wonderful illustrations of basic golden ratio concepts.

Definição / Definition (upper left): This shows the a line divided at its golden ratio. This is based on the ratios of the three points A, B and C. AB is created by the dividing point C, where AB/AC=AC/CB. This equation can be solved using the quadratic formula. The only positive solution is the golden ratio, also known as phi, and is calculated as (1+√5)/2.
Divisão do todo AB / Division of entire AB line (lower left): The geometric construction to divide a line at the golden ratio, where DB=AB/2.
Traçado de maior AC / Tracing of line AC (upper right): The geometric construction to extend a line to its golden ratio, where BG=CB/2.
Traçado de menor CB / Tracing of line AC (lower right): The geometric construction to extend a line to its golden ratio from its midpoint, where CF=AC and OC=AC/2

The sheet with four stamps shows:

Yellow: The rabbit population growth model identified by Leonardo Fibonacci. This led to his discovery of the sequence that bears his name.
Red: A pattern of alternating Fibonacci number spirals. These appear in the seed pods of flowers, pine cones and other plants. This particular example has 21 counter-clockwise spirals in green and 34 clockwise spirals in gold.
Blue: A pattern illustrating five-fold symmetry based on the solution identified by Roger Penrose. This is found in his phi-based “kites” and “darts.” These are called Penrose tiles.
Green: A Fibonacci spiral formed from squares of the successive Fibonacci numbers 1, 1, 2, 3, 5, 8 and 13. A spiral arc connects the corners of each square. A shell pattern with chambers based on a second spiral (the thinner line) illustrates the relationship to spirals in sea shells.

Use of the golden ratio in the design of the Macao stamps

Here too the designer used the golden ratio in the design of the stamp itself:

Left: The width of the stamp is based on the golden ratio of the width digits of Phi shown on the sheet.
Middle: The width of the golden spiral is based on the width of the digits of Phi shown in the white area.
Right: The width of the Penrose tiles is based on the width of the digits of Phi shown in the white area. In this case, it is at the golden ratio of the primary golden ratio.

Monaco 2000 Commemorative International Year of Mathematics Stamp

Monaco issued a 6.50 Franc / 0.99 Euro postage stamp in 2000 titled “Année internationale des mathématiques” (International Year of Mathematics) among their stamps issued for that year.

The stamp features Leonardo da Vinci’s famous illustration Vitruvian Man, which is based on the mathematical proportions of human design by Vitruvius.

While it covers aspects of mathematics beyond the golden ratio, it includes:

A golden ratio construction, with the note that dimension AK = 1.618 and φ. This would be in relation to the diameter HK of the circle.
A pentagon, and five-pointed star, the dimensions of which are all related to the golden ratio.
A dodecahedron, whose dimensions and the xyz positions of its vertices are all related to the golden ratio.

It appears that the stamp designer, Claude Andréotto, may have used the golden ratio in several aspects of his composition of the stamps:

Italy 1994 Commemorative Luca Pacioli Stamp

Italy issued a 750-lira postage stamp in 1994 to commemorate the 500th anniversary of the publication of Luca Pacioli’s Summa de Arithmetica. The inscription on the stamp “Della Summa de arithmetica, geometria, proportioni et proportionalita” means “The Summary of arithmetic, geometry, proportions and proportionality.” This book on mathematics was written by Luca Pacioli and first published in 1494. It contains a comprehensive summary of Renaissance mathematics, including practical arithmetic, basic algebra, basic geometry and accounting. Pacioli continued his work in this area with a study of the golden ratio in his publication of “De Divina Proportione”, or “The Divine Proportion” in 1509. While this stamp specifically relates to Pacioli rather than the Divine Proportion itself, it references his earlier work on proportions and proportionality.

San Marino 1992 Commemorative Divine Proportion Stamp

The Republic of San Marino, an enclaved microstate surrounded by Italy, issued a sheet of stamps in 1992 commemorating the life of Piero della Francesca. Much of his work was later absorbed into the writing of others, notably Luca Pacioli. Piero’s work on solid geometry appears in Pacioli’s “De Divina Proportione”, which has illustrations by Leonardo da Vinci. This work is recognized in the middle stamp of the set, which is captioned “de Divina Proportione”:

Switzerland 1987 Commemorative Golden Rectangle Design Stamp

Switzerland issued a stamp in 1987 to commemorate the 150th anniversary of the Swiss Society of Engineers and Architects.

It includes references to:

Helvetia, the female national personification of Switzerland, and also its name in Latin.
Stephan Bundi, a Swiss poster designer and graphic designer.
Jules-Ami Courvoisier, a Swiss designer who often used black-and-white effects or just two colors to create an expressive, figurative design. In addition to numerous posters, he also created stamps for the Pro Juventute and illustrated books.
The Schweizerische Ingenieur und Architektenverein (SIA), or Swiss Society of Engineers and Architects.
And last, but not least, the connection of the golden ratio to design. This was done with a golden rectangle spiral, with an arc connecting each 90 degree arc of its corners.

Dominica 1986 Commemorative Leonardo Fibonacci Stamp

The Commonwealth of Dominica, is a sovereign island country and part of the Windward islands in the Lesser Antilles archipelago of the Caribbean Sea. It issued a stamp in 1986 to commemorate Leonardo Fibonacci, with the Greek letter Phi representing the golden ratio.

Japan 1986 Commemorative Golden Section Stamp

Japan issued the “Golden Ratio Centenary of the Architecture” stamp in 1986 to commemorate the 100 year anniversary of the founding of Architectural Institute of Japan. It too recognized the relationship of the golden ratio to design. It did so with its illustration of the classic construction of a golden ratio using an arc swung from the midpoint of a square.

In addition to the golden ratio construction, the designer of this stamp very creatively incorporated the golden ratio into the design of a number of its elements. This is shown by the red grid lines, each of which is in golden ratio proportion to the ones on either side of it:

Nichtsburg & Zilchstadt 2004 Fibonacci Commemorative Stamp

In 2004, the non-existent nation of Nichtsburg & Zilchstadt issued a stamp bearing the image of Leonardo Fibonacci. This falls in the realm of fantasy issues, also referred to as “Cinderellas.” This stamp was included in a set of four stamps on mathematics that were created by Inzander Berenku, of Occussi-Ambeno’s Imperial Government Printing Office (KDPN). “Nicht” and “Zilch” mean “nothing,” so it was quite appropriate that the stamps commemorated mathematicians who advanced the concept of zero, or nothing. The 30 Midenika stamp honors Leonardo Pisano Fibonacci. Fibonacci brought nothing, or zero more precisely, to Western Europe from the Arabian world. Their decimal place-value numeral system quickly replaced the Roman numeral system. The “Midenika” is the plural of Miden, the modern Greek word for “zero”, and is a fictitious monetary unit. That’s a whole lot of nothing, isn’t it!

What about the United States Postal Service and other major countries?

You may have noticed that the world’s largest countries are absent from the list of those commemorating the golden ratio in their postage stamps. In fact, some of the countries that issued commemorative stamps are among the very smallest in the world, and one is even non-existent. Perhaps that’s because these small countries do not have their own rock stars, movie stars and super heroes to honor with a stamp. Yes, perhaps that’s the reason. On the other hand, maybe the pop-icon cultures of major countries just keep them from fully appreciating the finer things in life, like the golden ratio, that are truly timeless in their beauty and impact!

25 most popular stamps in the US

With a few exceptions, there’s a striking contrast in the subjects of stamps that are the most popular in American culture. Here’s the list of the 25 most collected stamps issued by the United States Post Office:

Rank	Subject	Year	Millions Collected
1	Elvis	1993	124.1
2	Wonders of America	2006	87.5
3	Wildflowers	1992	76.1
4	Rock & Roll/Rhythm & Blues	1993	75.9
5	DC Comics Super Heroes	2006	73.0
6	Greetings From America	2002	71.4
7	Insects and Spiders	1999	61.0
8	Art of Disney: Romance	2006	57.2
9	Legends of Baseball	2000	53.9
10	Art of Disney: Celebration	2005	52.8
11	Baseball Playing Fields	2001	49.0
12	Art of Disney: Friendship	2004	47.7
13	Civil War	1995	46.6
14	Legends of the West	1994	46.4
15	Marilyn Monroe	1995	46.4
16	Advances in Aviation	2005	45.8
17	Bugs Bunny	1997	44.1
18	Lunar New Year SS	2005	42.7
19	Comic Strip Classics	1995	40.8
20	Jim Henson: Muppets	2005	40.6
21	Cloudscapes	2004	40.1
22	Stars and Stripes	2000	40.0
23	Baseball Sluggers	2006	39.5
24	Summer Olympics	1992	39.5
25	Lucille Ball	2001	38.7

And here are the top 10 from the list:

Submitting ideas for new stamps

So how can we get more mathematically-oriented subjects on stamps? The Postal Service receives suggestions for new stamps from about 50,000 people every year. Only 20 to 25 of those make the cut. This selection is made by the Postmaster General’s Citizens’ Stamp Advisory Committee, which includes designers, historians and academics. They review the submissions and make their recommendations to the Postmaster General for final approval.

So there’s a lot of competition, but if you’d like to recommend that the Golden Ratio and Fibonacci series be recognized with a commemorative stamp in the good old USA, the Postal Service encourages those with ideas for new stamps to submit their suggestions in writing to:

Citizens’ Stamp Advisory Committee
Stamp Development
U.S. Postal Service
1735 North Lynn St Rm 5013
Arlington VA 22209-6432

Every ten years there is an international celebration of stamp collecting at the World Stamp Show, which was hosted in 2016 in New York City. The show is attended by stamp collectors, dealers, exhibitors, postal administrations and other societies from around the world. There they sell, buy and discuss all aspects of this hobby, which is known as philately. Maybe if enough of you reading this article write to the Postmaster General, we’ll see some more golden ratio commemorative stamps on display at the next show.

Create or purchase your own custom USPS postage stamps!

For those of you who just can’t wait that long or take the competition, there is still another option. You can create and order your very own custom US postage stamps at Zazzle.com. They’ve already created some beautiful Golden Ratio Stamps, and Fibonacci Stamps, including some of those shown here:

If you know of any Golden Ratio, Phi or Fibonacci stamps that weren’t covered in this article, please let me know and I’ll include them, with a credit to you. Send your submissions via the Contact page.