This is a story about a tool for telling apart two things that look the same, by moving something. It is fine if maths is hard for you. Take your time, and read along with the pictures.
If you are good at maths or physics, or if you want to learn more deeply or do research, please go to the detailed article “What Is a Separator?”.
A bird is flying across the sky. But is it really just one bird?
This is all you can see through the window. From the part you can see, you cannot tell whether there is one bird or two.
Migrating birds that meet, then part
In fact, there were two birds. On their journey the two met up and flew together for a while. If you only watch the part where they fly together, you cannot tell which bird is which.
Try moving the slider below to change the “day”. What happens when you move it forward? And what happens when you move it back?
The two birds overlap and look like one bird.
After the meeting place, the two birds went different ways. When you moved back, you saw that they had come from different places too. The two birds that looked the same had different roots. A root is where something comes from.
Here is one important thing. Even if the two birds had flown together right to the end, that would not prove they came from the same place. It would only mean “they did not split apart with this slider”.
The place names in the picture were made up for this story. It is not the birds who notice and tell them apart. It is the person who moves the slider.
Three words
Let us give names to what we have seen so far.
The thing you moved that made the two come apart.
In the words of the full article: for two things that look the same, what has to be moved before they behave differently. A separator is that thing. In Japanese, bunrishi.
They look the same, but they come from different places.
In the words of the full article: the number or the look is the same, but the source (the root) is different. In Japanese, bekkon.
They look separate, but they come from the same place.
In the words of the full article: two things met in different places share one source. In Japanese, dōkon.
In the bird story, the day of the journey was the separator. The two birds had distinct roots. Then how can we check for the same root? Let us look at that with a tree.
Checking a tree by its roots
1If we look only above the ground
It looks like one tree covered in flowers. The bottom of the trunk is hidden behind a hedge. Is it really just one tree?
2If we make the wind blow
When we made the wind blow, the branches on the two sides of the tree swayed separately. But the branches of one and the same tree also sway separately in the wind. They sway the same way whether there is one root or two. So the wind cannot tell us whether the roots are the same or different.
A tree draws up water through its roots and carries it all the way to the flowers at the tips of its branches. So we use water coloured with dye (dyed water). Only the flowers joined to a root that drinks the dyed water take on the colour.
3Pour dyed water on one side of the ground
Only the flowers on one side were dyed. Because only one side was dyed, these are two trees with different roots. This is a case of distinct roots. When we check the answer by showing what is under the ground, there are two roots.
A rule of the story: every flower joined to a root that drinks the dyed water gets dyed. In a real, large tree, the colour may not show up this clearly.
4If we try another tree
This is another tree with the same shape. We poured dyed water on only one side of the ground, yet the flowers on both sides were dyed. Even so, we can still only say “maybe the same root”. It could just be that the roots of two separate trees both reach the place where the dyed water is. We call this a candidate for the same root.
5Dig into the soil of that tree
When we dug into the soil, we found one root, joined to the branches on both sides. Only after following it this far can we say it is the same root.
The tree story teaches us three things. If only one side is dyed: distinct roots. If both sides are dyed: still only a “maybe” (a candidate for the same root). If you can follow it all the way to the root: the same root. Also, something like the wind, which makes even one tree move in separate parts, cannot tell them apart.
Two salmon in the same sea
Here is a made-up story. In the same sea, two salmon that look exactly alike are swimming side by side. People say that salmon go back to the river where they were born. In this story, when the river where a salmon was born gets more water, the salmon turns towards that river. Did these two salmon come from the same river?
1Two salmon swimming side by side in the same sea
Just by looking, you cannot tell which river each one came from.
2If we cool the sea
Both salmon dived deeper in the same way. The cold of the sea works on both of them in the same way. So this does not tell us.
3Raise the water level of the West River
Only the salmon from the West River turned. The other one did not move. The two salmon came from different rivers (distinct roots). Yet when we cooled the sea, the two of them dived together.
4This time, raise the water level of the East River
We put the West River back the way it was. This time, the other salmon turned. This one came from the East River. When we check with both rivers, it becomes clear where each salmon came from.
A rule of the story: the way the salmon answer the water level of the river where they were born is a rule made up for this story. Real salmon may not move like this. In the salmon story, the root is the river where each salmon was born. In the tree pictures the roots were at the bottom, but in the salmon pictures the roots (the rivers) are at the top. This story is borrowed from the note article “The same number can come from a different river” (in Japanese).
Words to remember
Try moving something.
Only one moves? Different roots.
Both move together? Still “maybe”.
The same thing happens in the middle of a calculation
This happens in maths and physics calculations too. When you work through a long, complicated calculation, perhaps anyone can lose track partway: what type of thing is the formula that just turned up, and what is it calculating? It is a mix-up similar to what computer programmers call a “type error”.
Suppose two amounts that came from different places meet in the middle of a calculation, as the same 2 or the same 4π (π, said “pi”, is the circle number 3.14…). Then you can no longer tell which is which. Even then, the words to remember are the same. Move one thing, and see whether only one of them moves.
For those who want to learn more deeply or do research
From here on, it is about mathematics and physics. The same idea is used to separate numbers such as 2 and 4π. It is fine to stop reading here.
The detailed article: What Is a Separator? Telling Two Look-Alikes Apart by Moving Something (the Moon and the Sun, the picture of two lines, six experiments you can move, and the list of papers)
This page explains, with simple stories, the ideas of papers that I published myself as preprints; they have not been peer reviewed. The place names in the pictures and the way the dyed water and the salmon behave are rules of the story.
All the papers are published under CC BY 4.0. The record of Paper 300, where the idea of the separator comes from, is on its YOI record page.