Fibonacci mod 9 · Pisano period calculator
Fibonacci Clock
Divide every Fibonacci number by 9 and keep the remainder, and the list repeats every 24 steps: 0 1 1 2 3 5 8 4 3 7 1 8 0 8 8 7 6 4 1 5 6 2 8 1. Write each 0 as 9 and you have the digital roots of the Fibonacci numbers from F(1) on, the famous 24 digit pattern. Any clock size repeats the same way, and the length of the loop is the Pisano period: 60 for the last digit, 300 for the last two.
- Period π(m)
- The loop
- Zeros per lap
- Opposites
- Repeats
- Famous
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How it works
Why the remainders go in circles
1. Fibonacci numbers
Start with 0 and 1. Each new number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and on forever. Divide one by the one before it and you get closer and closer to the golden ratio, 1.618...
2. Keep only the remainder
Pick a clock size m and divide each Fibonacci number by m. Keep the remainder, the way a clock keeps the hour. You never need the big numbers: add the last two remainders and take the remainder again. With m = 9, 5 + 8 = 13 gives 4, then 8 + 4 = 12 gives 3.
3. It has to come back
Each step depends only on the last two remainders, and there are only m × m possible pairs. So a pair must come back, and from then on everything repeats. The rule also runs backwards, so the first pair to come back is the starting pair, 0 then 1. The number of steps in one loop is the Pisano period π(m), named after Leonardo Pisano, the man we call Fibonacci. It is never longer than 6m.
The 24 digit wheel
The digital root of a positive number is its remainder after dividing by 9, with 0 written as 9. So the digital roots of the Fibonacci numbers follow the mod 9 clock: 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9, then again. Entries 12 steps apart add to 9, or are both 9. The 3s, 6s and 9s sit at every 4th place, so joined up they make a regular hexagon. The other 18 places hold only 1, 2, 4, 5, 7 and 8, the vortex loop digits.
Real vs legend
What is math and what is story
| Claim | Status |
|---|---|
| Fibonacci remainders repeat for every divisor m. The loop length is the Pisano period, named after Leonardo Pisano. Lagrange wrote about these repeats in 1774. | Real |
| Divided by 9 the period is 24, so the Fibonacci digital roots repeat every 24 numbers. | Real |
| The last digit repeats every 60, the last two digits every 300, the last three every 1500. | Real |
| The ratio of one Fibonacci number to the one before gets closer and closer to the golden ratio, 1.618... | Real |
| Spirals in many sunflower heads, pinecones and pineapples come in Fibonacci counts. | Real |
| Fibonacci numbers appear in every flower and plant. | Legend |
| The 24 digit pattern is a code of the universe, of time or of DNA. | Legend |
| The 24 digits prove the 24 hour day was designed. | Legend |
| The Parthenon or the ideal human face is built on the golden ratio. | No source |
Questions
Fibonacci clock FAQ
What is the 24 digit Fibonacci pattern?
Take the digital root of each Fibonacci number, starting from the first 1: 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9. After these 24 digits the list starts over and repeats forever. It repeats because the digital root only depends on the remainder after dividing by 9, and the Fibonacci remainders after dividing by 9 repeat every 24 steps.
What is a Pisano period?
Divide every Fibonacci number by the same number m and keep the remainders. The remainders always repeat, and the length of one loop is the Pisano period of m, written π(m). π(9) = 24, π(10) = 60, π(100) = 300 and π(1000) = 1500. The name comes from Leonardo Pisano, better known as Fibonacci.
Is the digital root the same as the remainder after dividing by 9?
Almost. For a positive whole number they match, except that a remainder of 0 is written as 9. So F(12) = 144 leaves remainder 0 and has digital root 9. The digital root of 0 itself is 0. That is why the wheel shows 9 where the remainders show 0.
Why does the last digit of a Fibonacci number repeat every 60?
The last digit is the remainder after dividing by 10, and each new last digit depends only on the two before it. There are only 100 possible pairs, so a pair has to come back, and the first one to come back is 0 then 1. For 10 that happens after 60 steps. The last two digits repeat every 300 and the last three every 1500.
How do I find the last digit of a huge Fibonacci number?
Divide its position by 60 and keep the remainder r. The last digit of F(n) is the last digit of F(r). For F(1000), 1000 = 16 × 60 + 40, and F(40) = 102334155 ends in 5, so F(1000) ends in 5 too. The look up field does this for any n up to 10^15 and any m.
Is the 24 pattern a code of time or the universe?
No source supports that. The 24 comes from dividing by 9. Divide by 10 and you get 60, divide by 11 and you get 10. The 24 hours in a day go back to Egyptian timekeeping, which split day and night into 12 parts each, and have nothing to do with Fibonacci numbers.
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