 ## The number TREE(3) is not infinite, but it is so big that it does not even fit in the universe

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there are numbers very very big. The one we bring you today is more than a trillion, more than a googol, more than a googoldduplex. And we are not talking about infinity. Neither of the gargantuan Graham’s number.

The number we are going to talk about is so large that cannot be written directly because the universe is not big enough. It is a number that goes beyond what physical laws can describe. A number so large that nature cannot fit their existence. And yet, mathematics allows it to be defined as a finite number.

## The immeasurable immensity of TREE(3)

TREE(3) is a finite number. But it is also a mathematical creation whose reach goes beyond physical limits. It is a nice example of how mathematics serves to describe reality, but nature does not always find representation for all mathematical entities.

Professor of Physics and Mathematics at the University of Nottingham, Anthony Padillahe explains on his excellent channel NumberphileHow does this number work? “Nothing compares to TREE(3),” explains Padilla. Here we leave you the original video for those who wish to hear the explanation directly from the teacher.

“It’s a number that appears easily just by playing a simple math game. However, it is so colossally large that it could not fit into our universe.” describes the teacher. But where does this number come from? As its name indicates (tree), the mathematical game is based on graph theory.

The objective of the game is to make a forest of trees, using seeds. That is, build branches based on different colors, which represent the seeds. The result of TREE(3) it’s so big we don’t even know how many digits it hasbut it has been shown mathematically that by definition it has a self-contained and therefore finite size.

We start with seeds, typologies or colors. Y we have two rules. The first is that the first tree can only have one seed; the second a maximum of two seeds, the third tree a maximum of three, the fourth a maximum of four, and so on. This is the creation rule.

The second rule is the conclusion rule. When one of these trees created contains a tree that had already been made before, the forest ends and the number is finished. In other words; if a previous form is repeated, it can no longer be followed. If we look at the following image we will see an example of repetition, since the green, black and red dots are the same and in the same orientation.

The TREE number is basically created following these two written rules. And this is where the beauty of the game appears. To understand how the number grows, let’s start with TREE(1), where there is only one seed, represented by the green dot. By making the first tree, we see that the result is just the same as the original and therefore the number ends with the first step. Result? Basically that TREE(1) = 1. In the case of TREE(1) the number ends early, since the first step repeats the shape. Image: Numberphile

Let’s continue with TREE(2). Here we have two seeds, which can be represented by a red dot and a green dot, for example. You start with a green dot like before, then build a tree with two red dots. Since it does not have any green dots, this tree is valid and the game continues. If in the third step we add a green point we would already be repeating. But there is an option and that is to put only a red dot. The rule says that the third tree can have a maximum of three seeds, but fewer are also allowed. Therefore the red dot is valid. And this is where the possibilities end. Result? TREE(2) = 3. TREE(2) also terminates fairly quickly, in just three steps. Image: Numberphile

And this is where the magic appears. While TREE(1) and TREE(2) are terminated very quickly. The game with TREE(3), that is, with three colored dots, is enormous. Colossal. boundless. We could spend a lifetime making combinations that we wouldn’t finish.

The mere incorporation of a third seed allows us to continue creating trees almost without end. “It’s much bigger than anything you can begin to imagine in physics,” Padilla explains. How big? unimaginable. Not even if we created a new tree in a Planck time and we had the age of the universe, it would not give us time to reach the end, points out the professor.

With two simple mathematical rules, a concept has been created that apparently has no end, but where it has been mathematically proven that we are not dealing with an infinite number. After all, it is a well-defined concept and the math suggests that eventually a tree will be found that contains a previous tree.

TREE(3) is so large that compared to even Graham’s number is practically zero. Although perhaps the most striking thing about this mathematical game is the huge jump from TREE(2) to TREE(3).

The mathematician Joseph Kruskal formulated the theorem about what any TREE(n) ends up resulting in a finite number. In other words, even if we used five colors (TREE(5), the result would be a non-infinite number. The only problem is that it is so big that not even in the lifetime of the universe could we reach it. It is a mathematical entity that goes beyond of the physical conception itself.An entity that shows how mathematics is used to describe reality, but sometimes there are concepts that do not have to be represented. This curious mathematical number reminds one of the reflections of the great physicist Richard Feynman in their classes:

“From our point of view, mathematics is not science, because the proof of its validity is not found in the experiment. Now, let’s clarify that if we say that something is not science, we are not saying that it has a problem; love neither It’s a science.”

Image | Nick Nice

there are numbers very very big. The one we bring you today is more than a trillion, more than a…

there are numbers very very big. The one we bring you today is more than a trillion, more than a…