Fourierflies

"What the hell is that?" I asked.  Something fluttered through the air, hard to focus on.  It had wings, but there seemed to be streamers or tendrils somehow attached to the wings that moved more slowly, slipping past, or perhaps through, things.  Dark shadows like cracks in reality.  Its wings were hazy and blurred, I got a sensation of greyness but there was nothing I say for sure that I'd seen.  It's body was somewhere between the wings, and I was even less sure I'd seen any of that.
"It's a Fourierfly," said Jacques, nodding his head to the beat of some music from his headphones.  The white cable stretched down his torso and disappeared into the pocket on his jeans.  I frowned and gestured at him, my hands flipping away from my ears.  He frowned back, and reluctantly removed his headphones.
"Thanks," I said, insincerely.  "What's a Fourierfly?"
"What you get when you apply a Fourier transform to a butterfly," said Jacques, deliberately unhelpful now I'd made him pay attention to me.
"I see."  I didn't.  "And why is an experimental creature loose in this room?  I know the red light wasn't on when I came in, and I'm also sure that you didn't have a form-12 stuck up outside either."
"Yeah, well...."  Jacques suddenly looked a lot more alert, aware that he had broken lab protocol and annoyed me.  He was probably worried I'd report him, and I was definitely considering it.  "They keep slipping out of their cages," he said.  I wondered if he was lying.
"Build better cages," I suggested.  "Ones that are Fourierfly proof."
"We're trying," he said, and now it sounded like I was getting closer to the truth.  He sounded worried at last.  "Only they keep slipping through the mesh, and we can't work out how they're doing it.  Well–" he sped up to stop me interrupting, "– we think we know how they're doing it, but we don't know how to stop it."
"You'd better tell me all about it then," I said.
"Well, the problem is the Gibb's phenomenon, we think," he said.  "It gives them nearly 10% wiggle room at the edges, and they keep using it.  They fly close to the mesh, and one time in ten they just pass straight through it.  It's kind of like quantum tunnelling, only on a macroscopic scale."
"Does this just apply to mesh?" I asked, feeling a familiar cold shiver run down my spine.
"Anything, really," said Jacques.
"Including the walls, doors, and windows of this room?"
"Oh yes.  Oh.... oh no."
"Do you count them?"
"Ye–... not often enough?"
"Probably not," I said.  "I wonder how many have escaped already.  But you still haven't explained to me what a Fourierfly is."
"It's the Fourier transform of a butterfly," said Jacques.  "You know that Professor Albert figured out how to perform a physical Fourier transform, and move things between Phase Space and Frequency Space?"
I just nodded, as Jacques appeared to have forgotten that I'd been Professor Albert's collaborator on that paper and had done most of the theoretical maths.  "He applies it to group of up to eight atoms, cooled to close to 0 Kelvin," I said.  "They have slightly odd properties when he does that, and his current research lab is trying to work out what all the properties are, and their ramifications."
"Yeah, well we did the same thing to a cage of butterflies for a bit of a joke," said Jacques.  His face clearly showed that he no longer thought it was a joke, and was pretty certain I wouldn't either.  "They all transformed."
"You weren't expecting that?"  Now I'd put him in a quandary.  Either he said he was, and looked careless and irresponsible, or he said he wasn't and looked like he didn't have the ethics to be trusted with lab animals.
"I was," he said eventually.  "But not all of them."  I nodded, it was probably the best answer he could give me at the moment.
"What did they transform into?"
"Still some kind of butterfly," he said.  "But they... I don't know how to describe it. They seem to have access to another dimension or something, so they can sometimes oscillate perpendicularly to... well, the real world.   And when they do, they can bypass things in their way."
That was the shadows I'd seen then.  Not fractures in reality, but new directions, new dimensions for light to fall into.  Of which we knew nothing, including if there was anything else in those dimensions, and if was safe to input energy, even in the form of light, into it.
"You'd better tell Professor Albert," I said.  "This is beyond us now, we need some seriously clever thinking applied to the problem."
Jacques looked as relieved as if I'd turned up with a reprieve five minutes before he was due a lethal injection.  "Thank-you," he said.  "Thank-you so much–"
"Yeah," I said, interrupting his frankly embarrassing effusion of thanks.  "Let's just get the mess sorted out and cleaned up before anyone else finds out about it."
I stood up and left, feeling an immense relief myself.  If Albert could find a way to undo the Fourier transform for the Fourierflies then perhaps I could use it myself and get my daughter back.