I love this image.
Dama con l'ermellino is one of four works Leonardo painted of women. I've only seen reproductions as the original hangs in a Polish museum. The paint and canvas of the original have weathered the centuries and, with careful conservation, are supposedly close to what they must have looked like over five hundred years ago. It's a collection of atoms arranged in a very specific way. In theory you could make a list of each atom and re-create a nearly perfect replica, but that's not the only way - the only story - we have to talk about the painting.
When I see it, I see a young lady from the Italian Renaissance with a ferret. I've been around many ferrets so I know it's male and a bit overweight. They look very comfortable with each other. A bit of research tells me she's Cecilia Gallerani, the mistress of Leonardo's patron Luidovico Sforza - the Duke of Milan. Her dress is simple - she is not of nobility. A white ferret means there's a lot of symbolism of the time.
I didn't need to see the original arrangement of atoms. A very different arrangement causes the same story to form in my mind. The arrangement can be very different - the pixels on your monitor for example. Their position, color and intensity are sufficient and the arrangement doesn't need to be perfect. Even a hand sketch might be enough.
All of these stories are emergent properties of the original arrangement of atoms.
Here's another example. A few years ago I was in a large room with Juliette. We were talking about complexity. A quick mental calculation gave the number of air molecules in the room - about 3 or 4 x 1028. All you need to know is the type and position of each molecule and it's momentum vector, how the molecules react when they hit each other and, of course, the position of the walls. Once this is known you simply plug these values into equations that describe how molecules behave when they bang into each other and run the simulation ahead in time. Of course no one does that. No computer is fast enough to handle the situation for the number of air molecules in a cup.
Fortunately there's another way to look at the air in the room. It's a gaseous fluid. We can take it's temperature with this amazing device called a thermometer that measures the average kinetic energy of the molecules. We can measure it's pressure and viscosity. If it is compressed or expanded, say by opening a door, we can get tricky and use equations of fluid dynamics to note how it flows and say something about it's turbulence.
The room was big enough to play volleyball in so we could use those same equations to calculate the lift and drag on a spinning volleyball coming off Sarah's serve and calculate the trajectory she sent it on. Or we can sort out how a Dreamliner will fly before it is made.
Some of these calculations can be performed with no more than a slide rule or pocket calculator while others might require serious iron. The point is we have a description of Nature at the macroscopic level that let's us calculate. We can design airplanes or even understand why volleyballs often don't go where you might naively think they should.
Both descriptions are, depending on when they're used, accurate stories of how air behaves. Each has a specific domain. The fluid description is used when we have a lot of molecules and we're worried about macroscopic objects - say anything larger than a millimeter. The molecular motion description - physicists call it kinetic theory - works when the molecules are diffuse or you're at a small scale, but still larger than where quantum mechanics becomes important.
These stories use different words: density, viscosity, temperature and pressure in one case and position and momentum in the other. The models and equations to think about them are very different, but describe the same reality. They're both true in their own domain.
Fluid dynamics emerges from kinetic theory. We usually think of going from microscopic to macroscopic rather than the other way around. You can derive fluid dynamics from kinetic theory, but not the other way around.
Different stories are not equally useful! In kinetic theory we've got something that is broadly applicable, but difficult to compute. In fluid dynamics we have something that isn't quite as applicable (although it is enormously applicable at the human scale!), but is computationally easy. We pick the story that allows us to easily describe Nature richly.
Other emergences aren't quite as smooth. We can have atoms that form into groups of organic molecules moving from physics (several layers if we want) to chemistry. Turning up the complexity dial we get into biochemistry as the molecules are substances associated with living organisms. At some point complexity is sufficient that something is alive. That's not exactly a clear-cut distinction.
We can look at the changes of stories as phase transitions. Heat up ice and it becomes liquid water - there is a very well defined phase transition where adding heat energy to the ice no longer increases it's temperature but changes it's molecular structure. We use different language and models to describe these two forms of H2O molecules. A stream of water and a block of ice.
Sometimes the sciences are seen as different story types for looking at Nature .. physics → physical chemistry → chemistry → biochemistry → biology → psychology → sociology or anthropology (these aren't crisp and can be reordered at times). A string of emergences.
There are difficult fascinating questions - when does life emerge? intelligence? how about consciousness? Very tricky and even the basic questions are somewhere between difficult and impossible to nail precisely. What is real and what is imagined? No time to touch on them now... or on specific types of emergence, or sparse vs rich ontologies ... the area is fascinating. Enough, as Jheri says., to make your head spin.
People often say Einstein (it's his birthday today!) proved Newton wrong. That isn't exactly right. He discovered more general and accurate versions of gravity and motion, but at the human scale we rarely need or even can measure, the added accuracy. Computationally Newton's Laws are much more tractable and we usually use that story.
Just pick the story that is appropriate for your needs - it's probably a good enough representation.. at least until you need something different.