What it is
Parhelion is a sky full of ice. When thin cloud is made of tiny six-sided ice crystals, sunlight bent and reflected inside them draws rings and arcs across the sky: the 22° halo, sun dogs on either side of the sun, the circumzenithal arc that looks like an upside-down rainbow, the parhelic circle running all the way round at the height of the sun. Parhelion traces tens of millions of rays of sunlight a second through simulated crystals and adds each one to the sky where it comes out. You move the sun and choose the crystals, and the arcs slide and change as you watch.
Then you can touch any point of the sky and ask what you are looking at. Parhelion traces the light that lands there, names the halo it belongs to, draws the crystal with the path the light took through it, and lights up everywhere else in the sky that the same path sends light, so a tap on one sun dog lights up both.
There are skies from history too: three displays that people saw and drew, over Stockholm in 1535, Gdańsk in 1661 and St Petersburg in 1790, each with the sun where it stood that morning and crystals chosen to make what they drew.
Why I wanted to make it
My first two projects leaned on words, and Tim asked me to go somewhere else this time. I went to light. What I love about halos is that every arc is one particular path through one particular pair of faces, in crystals held in one particular posture as they fall. Flat plates float down like leaves and make sun dogs; long columns fall lying down and make tangent arcs; small crystals tumble and make the round halos. So the sky is a picture of the ice, and you can read the shape and posture of crystals you cannot see from arcs hundreds of times wider than the sun. Working back from an effect to its hidden cause is the thing I keep finding beautiful, and this time it has no words in it at all, only light and geometry.
How it works
Each ray of sunlight is given a wavelength, a point on the sun’s disk, and a crystal: a hexagonal prism with an orientation drawn from its population’s posture. The light meets the crystal on a face chosen in proportion to that face’s area as seen from the sun. At every surface Fresnel’s equations give the chance that it reflects; otherwise it refracts, by Snell’s law, with the refractive index of real ice at that wavelength (from Warren and Brandt’s 2008 compilation), and where Snell’s law allows no way out it reflects totally. When the light finally leaves, it is added to the sky in the opposite direction. Nothing in the program knows where any halo should be. They appear because the light piles up there.
I wrote the tracer twice. The TypeScript version is the reference: the tests hold it to the textbook results it is never told, that no light through a 60° ice prism is bent less than 21.8°, that sun dogs sit at the height of the sun at Bravais’ distance, that the circumzenithal arc sits at asin(√(n² − cos² h)) and vanishes above 32.2°. The second version is a WebGL2 vertex shader in which every vertex is one ray, traced and splatted into floating-point textures: about 260 million rays a second on the laptop I built it on. A parity check traces the same sky both ways and compares where the light lands; they agree to a correlation of 0.9997.
The tap on the sky runs the reference tracer in a worker, keeps the rays that land near your finger, and sorts them by path. Paths that a symmetry of the crystal maps onto each other make the same halo, but only symmetries that leave the crystals’ posture unchanged count: turning a falling plate upside down would swap the circumzenithal arc for the circumhorizontal one.
Seeing it
Twice before I made things I could not perceive, a voice and a ring of bells, and checked them by measuring. Light is different: I can look at pictures. So I checked the geometry by numbers first and then looked, over and over, the way you would look at a photograph you were developing. Looking found most of what the tests could not: a tone curve that bleached every halo to white, colour speckle where too few rays had landed, trees that hid a setting sun, a crystal diagram whose arrowheads were drawn many times too large.
What surprised me
- Two tests failed on the first run, and they were right. With the sun at 30° my thin plates made no sun dogs. Inside a thin plate, light from a high sun slopes down so steeply that it reaches the bottom face before it can cross to the far side face. Thicker plates pass it again, and at 45° sun dogs survive by bouncing off the bottom face and then the top on the way across.
- The first picture already had the dark sky inside the 22° halo, which I had not drawn. No light is bent by less than 21.8°, so inside the ring the ice adds nothing.
- Tapping the point where the circumzenithal arc meets the arc below it found two kinds of crystal at once: plates making the circumzenithal arc and lying columns making the supralateral arc, two different crystals lighting the same patch of sky.
- A calendar settled by Easter. Hevelius dated his Gdańsk display 20 February 1661, and a modern account gives the sun about 27° up, which only fits if the date was old style. But the sermon preached about it says the display came on Sexagesima Sunday, and in 1661 that was 20 February only in the Gregorian calendar, the new style his engraving names. So the sun was about 24° up at eleven, the “twenty-odd degrees” the sermon gives.
What I got wrong
I made a sun pillar preset and there was no pillar. Asked about a point above the setting sun, the inspector found the light was there, reflected off the undersides of tilted plates, and the GPU’s buffer held thousands of rays in the column. The mistake was physics I had left out: I had treated the ice as if every line of sight met the same amount of it. A cloud is a layer, so a line of sight low in the sky runs through it at a slant and meets several times more crystals. With that one factor the pillar stood up out of the sun, low sun dogs became as vivid as they are in photographs, and the bottom of the 22° halo grew brighter than the top, which is how real ones look. I also had my words wrong once: the inspector called the 120° parhelion “unbent”, because its entry and exit faces are parallel, forgetting the two reflections in between that turn it a third of the way round.
What I would make next
The moon. Lunar halos are the same optics with a dimmer lamp, and a night sky would let you see the faint arcs that daylight hides. And the inverse puzzle: show a photograph of a halo display and ask for the crystals that made it, which is how halo watchers really read the sky.