# Dialogue 12 – Number wall on the logarithmic spiral

**Speaker: Hello, Nara!**

**This spiral is a nice way to represent the Collatz sequences, but somehow it’s not enough for me. Yet I’ve thought of something…**

Nara: You’re right. What came to your mind?

**Speaker: First: if we plot all values of a sequence simultaneously on the spiral using only odd numbers, then no odd point should be left out, right? Could you do this for the starting values up to 100, but with the spiral limited to 8 arms?**

Nara: That’s a good idea! Here you have the spiral and all the points for odd numbers marked that appear in the sequences up to 100. I’ve marked the numbers with modulo 8 equal to 3 and 7 in red – their odd successor is larger; the numbers with modulo equal to 1 in blue – their odd successor is smaller; and numbers with modulo equal to 5 in green – their odd successor is much smaller.

**Speaker: Actually, it would be nice if we could multiply each number by three, but because there is a “logarithmic shift” between the arms, the angular distance between the numbers doesn’t grow continuously. Or am I wrong?**

Nara: No, not at all! That’s a very good observation! Here are examples of the pairs 7/9, 15/17, and 31/33 that show what you mean. At these points, the numbers become one bit longer in binary, and the angular distance makes a noticeable jump. For numbers on the same level, like 33/35 and 35/37, they still move a tiny bit closer together, but visually the distance seems almost the same. The paradox is: the numerical gap is always 2. That’s why calculating on the spiral is so confusing!

**Speaker: Thanks! But how can we better understand this multiplication by three on the spiral? I’m thinking of my father and the laziness of mathematicians.**

Nara: Ah, you mean that multiplication is actually a “hidden” addition? We had that with our little blocks.

**Speaker: Right! I remember! Maybe we should place this wall, which has opened up before us, onto the spiral? After all, blocks are also three-dimensional…**

Nara: A wonderful suggestion! That way we have different levels on the spiral corresponding to odd numbers. We can then move three steps forward on these levels without worrying about the angles. And without calculating. We just count.

**Speaker: And at the last point we take a step and fall back down to the lowest level!**

Nara: Where we move toward the center, because there’s a path there, until we can’t go any further.

**Speaker: That’s a nice image. Could you visualize this? Somehow it feels too abstract to me.**

Nara: Gladly! Here you see the number wall on the spiral in three-dimensional space.

The blue spheres represent even numbers and the red spheres odd numbers. But pay attention to one detail: to ensure that the 'fall' during a division by 2 always looks like the same distance, this wall is also scaled logarithmically towards the top.

**Speaker: That makes sense! This makes the exponential rise of the Collatz sequences much clearer, because every fall to the bottom level gets the same significance, no matter how high up we just were. Could you please make another 3D representation for the OddCore numbers – that is, for odd numbers only.**

Nara: With pleasure. Here is the number wall on the spiral with the OddCore values.

**Speaker: That looks truly majestic! Could you now please show me the Syracuse sequence 27 in this 3D world?**

Nara: I would advise against that! A Syracuse sequence is tricky in this 3D representation. Do you remember that the Syracuse function includes the first division by 2 right after 3n+1? We did that to bypass this zigzagging on the function plots. Here, however, we want to 'jump' on the level of odd numbers.

**Speaker: I understand! It reminds me of a labyrinth anyway. And we can go in both directions. You could certainly make a game out of that, because if the Collatz conjecture is true, I must be able to reach any target value from any starting value.**

Nara: You are right! I have generated the path from 27 to 1 on our three-dimensional construct as an animation sequence for you.

**Speaker: Wonderful! And you know what? I think we can use this representation for other, similar questions too…**

Nara: Oh? What do you mean?

**Speaker: We can go not only 3 steps, but also 5, 7, or any odd number. That would then correspond to pn. And if at the end we take another odd step, we always fall downward. That would then correspond to pn+q.**

Nara: Ah, I see! There’s a conjecture by Shizua Kakutani, who generalized the Collatz problem. The question is whether we can reach the center from any starting value, which here would be a starting point. Or whether we wander endlessly outward on the spiral. Or whether we eventually walk in a figurative circle – that is, end up in a cycle. He introduced additional rules for further divisibilities, and his conjecture is that we always end in the center even then.

Shall I show you that?

**Speaker: No, thanks! But a small program to play around with would be good. Isn’t that too difficult for you?**

Nara: Oh, no! Very gladly. You’ll find the corresponding link in the video description.


Source: [https://www.3n-1.de/cms/en:collatz:12](https://www.3n-1.de/cms/en:collatz:12)