# Dialogue 13 – Logarithmic spiral and q (at pn+q)

**Speaker: Hello, Nara!**

**This spiral fascinates me somehow! Surely there’s more to discover with it? I mean, if we leave out the third dimension…**

Nara: I’m glad you like it! What are you thinking of?

**Speaker: For example, the distance between the spiral arms. We can define it arbitrarily, and it always corresponds to either a multiplication by 2 — then it’s a jump to the next outer arm; or a division by 2 — then it’s a jump to the next inner arm. And it doesn’t matter how far apart the arms actually are geometrically, does it?**

Nara: You’ve understood that perfectly! This allows us to overcome any distance. For the Collatz function, geometrically only the angle between two odd numbers matters. In principle, we could even fix the spiral radius and obtain a circle on which the odd numbers are distributed.

**Speaker: Wow! But let’s stay with a spiral for now. That means I can multiply all numbers by a factor, right? Let’s take factor 5, for example. Then for each number on the spiral we have 5n**

Nara: Oh, I see what you mean! And yes, that works. We’ve had this before. Do you remember when you asked me whether the value of the coins in Egyptian multiplication mattered? That’s exactly what’s happening here.

**Speaker: So it doesn’t matter at all which factor I use? Even “crooked” numbers?**

Nara: Yes, exactly! By “crooked numbers” you mean fractions — those are relative numbers, and numbers that can’t be represented by fractions are irrational numbers.

**Speaker: But what happens then with parity? That only exists for whole numbers, right?**

Nara: Correct, that’s a very good point! Whole numbers are natural numbers. But since we’re dealing with a factor, we simply convert the “crooked number” on the coin back into a natural number. And then we can distinguish parity again. Parity depends on the number of coins, not on their value.

Look, if we take the factor 5, as you suggested earlier, we just need to set a slight precondition for the Collatz function. We write this 5 on every coin. Therefore, we first have to multiply the starting number by 5. And then we divide each result by 5, and from that we can read the parity. However, instead of 5, we can use whatever we want. The scaling factor is the value of the coin.

**Speaker: You mean, even even numbers?**

Nara: Yes, even even numbers! And not only even numbers, but also negative numbers.

**Speaker: Wait! That means 3n+1 becomes 3n-1…**

Nara: Yes and no. Our function now looks a little different: We prepare our starting value. It is an arbitrary natural number and corresponds to the number of coins. Let’s take 7.

First we multiply this 7 by our stretching factor — let’s say minus 2. So we start with minus 14.

And then our function is:

If the result of our calculation divided by our stretching factor is even, we divide by 2; in our case: minus 14 divided by minus 2 is 7 — our actual starting value, but that’s odd. So we take the other rule:

If the result of our calculation divided by our stretching factor is odd, we multiply by 3 and add our stretching factor. So we calculate minus 14 times 3, which is minus 42, and then add minus 2, our stretching factor. That gives minus 44.

Now the game begins again, but without changing our current starting value! We check whether minus 44 divided by our stretching factor minus 2 is even or odd. Minus 44 divided by minus 2 is 22 — even…

**Speaker: Oh, could you show this with a few graphs? Just with numbers it’s pure horror to imagine!**

Nara: Of course! Here’s your favorite Syracuse sequence 27 in a logarithmic representation. In red is the original function 3n+1. For the other functions I’ve written 3n+q in each color. What do you notice?

**Speaker: The graphs look the same and are only shifted on the y-axis!**

Nara: Exactly!

**Speaker: Sorry, but this is getting a bit much. Let’s continue later?**

Nara: Yes, you’re right. We have time and don’t need to understand everything at once. If you like, you can try out what we’ve discussed yourself. The link is in the video description.


Source: [https://www.3n-1.de/cms/en:collatz:13](https://www.3n-1.de/cms/en:collatz:13)