This white paper sets forth the official Golden Ratio Day (GRD), commonly referred to as Phi Day.

For the purpose of this paper the Golden Ratio will be represented by the 21st letter of the Greek alphabet, lowercase phi, f, algebraically ½(1+ Ö5), approximately 1.61803398874989484820458683436....

The uppercase Phi (F) is the Golden Ratio conjugate, 1/f, which is approximately 0.618033988749894....

Pi Day is an established date, set on March 14th. This is because the decimal representation of pi (p) starts with 3.14..., hence 3/14, March 14th.

Previously, the established Phi Day, or Golden Ratio Day (GRD), appears to be June 18th, based on the number 0.6180339.... This was presumably chosen because f is elegantly enough 1 more than its reciprocal, as f = 1 + (1/f). Therefore 1/f, or F, has the same decimal non-repeating irrational form, but doesn’t include the leading 1 to the left of the decimal. This yields 0.618..., the first three digits of which apply nicely to the Gregorian calendar as 6/18, June 18th.

To include the leading 1 would give us 1.618..., the only reasonable calendar representation being January 6th, a number perhaps too close to some civilization’s major holidays to be satisfyingly of interest.

Neither date is acceptable, as the adopted June 18th is inaccurate and both miss proper representation of the essence of the Golden Ratio, which is to divide a line into mean and extreme ratio.

In the Northern Hemisphere Phi Day (GRD) falls on October 31st, dependant on the Spring Equinox.

In the Southern Hemisphere Phi Day (GRD) falls on May 6th, dependant on the Spring Equinox.

Rather than compromising on a decimal form, Phi Day is based on applying f to the Earth’s natural yearly cycle. f is the technique by which to divide a line into mean and extreme ratio segments. So if we take Earth’s annual year to be roughly 365 days, and apply the Golden Ratio to it, we get approximately 225.582405.... Starting January 1st, 225 days later brings us to August 13th. This at first seemed reasonable.

But let us consider further – why January 1st? Although the Romans began the calendar year on January 1st starting in 153 B.C., the beginning of the year has moved around a lot since. It finally went back to January 1st in some European countries as early as 1522. The Gregorian calendar superseded the Julian calendar in 1582, and rose to become the most popular calendar on Earth, as it is today.

But starting the calendar year on January 1st to derive the Golden Ratio day is to settle for convention. The GRD should not be based on adopted custom, so we must choose a more pivotal point at which to start the Golden Ratio Year.

For Phi Day, the year’s most logical starting point will be adopted. This is to coincide with the natural progression of the Earth’s seasonal framework, and we'll start with the Northern Hemisphere as it has twice as much land as the Southern Hemisphere, and 9 times the population, but we will provide a date for the Southern Hemisphere here as well.

The Earth goes through a complete seasonal cycle with each revolution around the star at its solar system’s center. For most of the human occupants of planet Earth, there is typically a division of that cycle into four (4) seasons. They are Spring, Summer, Fall, and Winter. During Winter, nature, in much of its vegetative form and some animal forms, goes into dormancy. Much of the world is cold, covered with snow and ice, and historically life was difficult as shelter was extremely important, and all food to be consumed had to be previously harvested and stored for those months when not much vegetation grows. To the ancient people living on planet Earth, the end of Winter - the changing of the season into Spring - would have been a joyous time of celebration, as that marked the end of a “dead” period and a “birth” into a new year of growth and hopeful bounty. Therefore the most logical day upon which to build the year is what we call the Spring (Vernal) Equinox, March 20th.

Starting on March 20th (March 21st in 2007 and occasionally that date in Asia, but March 20th for the next century in North America) and dividing the ~365 days in a year into mean and extreme ratio, that is, the Golden Ratio, we need to apply an offset of 225 to 79 (March 20th), which brings us to the 304th day of the Gregorian calendar year, which is October 31st.

September 23rd is the Vernal Equinox in the Southern Hemisphere – note that the seasons do not consist of equal number of days because the Earth’s speed around the sun is variable due to its orbit - so Phi Day south of the equator falls on May 6th. It would be nice if this were May 1st, but it would be inaccurate to shift GRD to coincide with traditional holidays like Beltane (which in the Southern Hemisphere is celebrated on November 1st anyway) or Samhain, although GRD in the Northern Hemisphere happens to fall on October 31st.

In the Northern Hemisphere Phi Day is October 31st.

In the Southern Hemisphere Phi Day is May 6th.

Please feel free to email me your comments and criticisms. Whether you choose to celebrate on October 31st, June 18th, or any other day, we are all celebrating one of mathematics most beautiful, challenging and satisfying constants.