“As we as architects begin to think about designing not an object, but a process inspired by nature to generate objects, in short we will have no constraints. With the processes in our hands that allow us to create structures that we couldn’t even have dreamt of. And at one point we will build them.” — Michael Hansmeyer, TEDGlobal 2012

]]>https://johncarlosbaez.wordpress.com/2015/08/11/the-physics-of-butterfly-wings/

Complex! But make it simpler. Get the image up on your monitor, tweak it in whatever way you can until you like it. Now paint from it. I still use a printed image but I compare it to my monitor and paint accordingly. When I take a picture of the painting I adjust it until it looks like the painting. I still can’t guarantee that what you see on your monitor will match the painting exactly but no matter what I do I can’t guarantee that.

]]>Interestingly, it says butterfly wings may take advantage of topological defects. The paper is freely available here:

• Andrej Singer *et al*, Domain morphology, boundaries, and topological defects in biophotonic gyroid nanostructures of butterfly wing scales, *Science Advances* **2** (10 June 2016), e1600149.

Unfortunately it’s a bit short on clear explanations of what they found.

]]>New article about butterfly wing colors

]]>• The physics of butterfly wings.

It turns out that the pattern of a gyroid is closely connected to the triamond:

• Matthias Weber, The gyroids: algorithmic geometry III, *The Inner Frame*, 23 October 2015.

Instead of trying to explain it here, I’ll refer you to the wonderful pictures at Weber’s blog.

]]>Bruce wrote:

Can you elaborate on that (in the simple classical case)? It doesn’t sound like any classical mechanics I know….

Thanks, you caught a mistake. I should have said:

If you have a charged particle in a

magneticfield, its momentum is not its mass times its velocity.

and I’m actually going to go back and change that word in my earlier comment.

First, note something more familiar. The energy of a charged particle in a electric field is not just there’s a correction due to the scalar potential:

where is its charge and is the scalar potential at the particle’s location. The scalar potential, also known as the electrostatic potential, is the field with

where is the electric field.

It turns out that in a similar way, the momentum of a charged particle in a magnetic field is not just there’s a correction due to the vector potential:

where is the speed of light and is the vector potential. The vector potential is the field with

where is the magnetic field.

All this is nicely explained here.

]]>“If you have a charged particle in an electric field, its momentum is not its mass times its velocity. ….”

Can you elaborate on that (in the simple classical case)? It doesn’t sound like any classical mechanics I know….

]]>Arch1 wrote:

John, I was indeed assuming that momentum is proportional to speed.

Okay. Yeah, when you first start learning physics, ‘momentum’ seems like a fancy name for mass times velocity, perhaps invented by lazy physicists who prefer to write one letter instead of two.

And once upon a time, maybe people thought that’s all momentum was. But it took on a life of its own and now we realize it’s quite different. If you have a charged particle in a magnetic field, its momentum is not its mass times its velocity. And if you have an ordinary particle and take special relativity into account, its momentum is not just its rest mass times its velocity.

So what is momentum? The real definition of momentum is perhaps easier for waves than for particles. If you have a wave like

where is time and is the position vector, then its energy is and its momentum is the vector

So, *energy says how rapidly a wave oscillates as you move forwards in time, while momentum in any spatial direction says how fast the wave oscillates as you move in that spatial direction.*

That’s all there is to it, nowadays.

]]>John, I was indeed assuming that momentum is proportional to speed. I’ll read your remedial instruction (thanks) and follow up on the links.

Re: my joke, I get that a lot. (I’m beginning to think my wife is right about my sense of humor:-) In this case, the intended humor is that the thing I labeled a “subtle regularity” is (I think) just a consequence of how the .GIF was built.

I *assume* the .GIF shows the gyroid being built up layer by layer, as if the gyroid were being printed by a 3D printer. In which case I *think* that the surface normal at the points of joining (the points at which two separate pieces of geoid first touch) must be perpendicular to the planes defining the layers that are being added.