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CHAPTER 17
Friends and dragons on the gargoyled eaves, Watch the dead Christ between the living thieves. —Longfellow, Divina Comm., Sonnet ii, 1864
Out of the Mouthes of certain beastes or gargels did runne red, white, and claret wine.” —Chronicles, 1548 A rusty iron chute on wooden legs came flying, like a monstrous gargoyle, across theparapet. —Stevenson, Silverado, 1883 |
There is a closeness to everything: the shiny stalactites behind you blending into the darkness, the dripping of water, the dense shadows of the stalagmites. “Beautiful,” you whisper.
You are surrounded on all sides by deep galleries of quartz rising to support the cavern roof. You gaze up at tiny amber stalactites that cover much of the roof, beautifully arched. Tears almost come to your eyes.
Theano smiles and nods. “Incredible,” she says.
You feel goosebumps rise along your arms as you look at Theano. “I, I wish I had met you sooner, before the transfinites appeared, before I became preoccupied with the End of the World, before I knew a comet would someday hit the Earth.” You stare into her lovely eyes which bioluminesce like fireflies in the shadows of the stalactites.
The silvery light from the flashlight is falling on her brown hair. Her eyes are moist. How beautiful, yet sad, she seems.
“We’re together now,” she says as she presses her hand into yours. She hugs you, running her fingers up through your hair.
Mr. Plex coughs. “Sir, what is our lesson today?”
You take a deep breath. “Gargoyles.”
Historical gargoyles from the Cathedral of Notre Dame, Paris.
As Mr. Plex squats down to listen, his joints make a peculiar crunching sound like the breaking of bones.
You reach behind a stalagmite and withdraw a stone creature resembling a devil. “Got this from the Cathedral of Notre Dame in Paris.”
Theano gasps.
You run your hand along the rough stone. “The term gargoyle originally referred to grotesque water spouts shaped like bizarre animals or monsters. They projected from the gutters of buildings, especially in Gothic architecture, carrying the rain-water clear of the building. Today they also refer to grotesque masks and other ghostlike visages.”
“Sir, what’s this have to do with mathematics?”
“We’re going to talk about numerical gargoyles. They’re based on a cellular automaton consisting of a grid of cells which can exist in two states, occupied or unoccupied. The occupancy of a cell is determined from a simple mathematical analysis of the occupancy of neighbor cells.” You pause as you stretch your arms. “Though the rules are simple, the patterns are very complicated and sometimes seem almost random, like a turbulent fluid flow or the output of a cryptographic system.”
“Sir, can we program this on a computer?”
You nod and withdraw a notebook computer from behind a stalagmite. “I’ve put the program in the machine already. Just turn it on.”
Mr. Plex grabs the computer from you, flicks on the switch, and in seconds is mesmerized by the undulating and ghostly physiognomies. “Amazing,” he says. “It’s hard to believe this comes from such simple rules in a checkerboard world.”
Numerical gargoyle produced by a mathematical simulation.
You nod. “Look, there’s a weird one.” You point to a small evolving blob which appears to have two hollow eyes and a screaming mouth. “The space is just teaming with numerical gargoyles. You can use the computer program like a microscope to magnify interesting regions.”
“Wild,” Theano says.
The reflections from the computer screen make the cave walls pulse like a living lava lamp. A series of pastel circles moves around the walls of the caves, as in a patternless kaleidoscope. Slowly they coalesce into a beautiful blob of bathybius.
“Let’s stretch our legs,” Theano says.
The three of you get up and walk for a few minutes beneath an inverted forest of bones—ivory stalactites which form the tunnel. A few delicate helicites grow like fungus on the walls. Gargoyles
Theano suddenly cocks her head. “What’s that sound?”
You hear wind whistling through the cave and smell a vague scent of ammonia. It feels as if the cave is breathing, inhaling and exhaling great quantities of air.
You turn to Theano. “Nothing to worry about. Just the wind. I don’t—” you stop yourself suddenly.
Theano looks into your eyes. “What’s wrong?”
You stoop down. “Look at that. A footprint—”
“Gargoyle!” Theano screams with a quick indrawn breath.
“What?” You spin around and see two feeding tubes shooting out from your Notre Dame gargoyle. The tubes slowly trail along the wet cave floor. Red, engorged arteries are braided along the tubes’ entire length.
You clench your fists. “Those transfinites. They’re trying to scare us again.” You turn your face to the ceiling of the cave. “We’ve done nothing wrong,” you say to no one in particular. “We’vedone our best not to violate the Main Directive. We’re not going to areas of human habitation anymore. We only want to learn.”
Theano runs toward you. “What do we do now?”
Mr. Plex, as quick and quiet as a ferret, runs up to the gargoyle. He stares at it for a second, then raises up his body like a cobra about to strike, and smashes the gargoyle with his right forelimb.
Numerical gargoyle produced by a mathematical simulation
“Oh, Gargoyle, darling,” she said, sitting down on an old hitching block at the edge of the Rosedale pavement, “isn’t it too gorgeous?”
—J. Harris, Weird World Wes Beattie, 1964
The term “gargoylism” has been applied to this syndrome because the gross disfiguration resembles the gargoyles of Gothic architecture.
—R. Durham, Encylcopedia of Medical Syndromes, 1965
Gargoyles and Gargoylism
Originally the term “gargoyle” referred only to the carved lions of classic cornices or to terracotta spouts, such as those found frequently in Pompeii. The word later became restricted primarily to the grotesque, carved spouts of the Middle Ages. The gargoyle of the Gothic period is usually a grotesque bird or beast. Today, the term “gargoyle” has gradually evolved to include other morbid beasts such as chimeras decorating the parapets of the church of Notre Dame in Paris.
Gargoylism is a syndrome characterized by mental deficiency and skeletal deformities including an abnormally large head, short limbs, and protruding abdomen. The disease is also known has Hurler’s syndrome.
Graphical Simulation
Numerical gargoyles undulating through horror movies of the future will include computerized versions where the ghostly faces are mathematical forms displayed on a computer screen. In order to create the monstrous faces, the simulation produces shapes that move, coalesce, and break up on a 2-D grid. The simulation involves the use of “cellular automata.” Cellular automata (CA) are a class of simple mathematical systems which are becoming important as models for a variety of physical processes. CA are mathematical idealizations of physical systems in which space and time are discrete. As you told Mr. Plex and Theano, CA usually consist of a grid of cells which can exist in two states, occupied or unoccupied. The occupancy of one cell is determined from a simple mathematical analysis of the occupancy of neighbor cells.
To create a CA, each cell of the array must be in one of the allowed states. The rules that determine how the states of its cells change with time are what determine the CA behavior. There are an infinite number of possible CA, each like a checkerboard world. I produced the shapes for the figures on pp. 182 and 183 by initially filling the CA array with random Is and Os. The next section discusses the rules of growth that determine the states of the cells in subsequent generations.
Grow Your Own Gargoyles
The gargoyles in this chapter evolve in discrete time according to a local law. Specifically, the value taken by a cell at time t + 1 is determined by the values of its neighbors (and its own value) assumed at time t:
In this equation, 
denotes the state occupied at time t by the site (i,j). The nine-cell template I used to produce the numerical gargoyles is called a Moore neighborhood (as opposed to a von Neumann neighborhood consisting only of orthogonally adjacent neighbors). For example, let’s denote the current cell and its position i (horizontal position) and j (vertical position) in the checkerboard by c(i,j). Its neighbor cells are c(i+l,j), c(z – l, j), c(i,j+1), and c(i,j – 1) for the cells up, down, right, and left of the current cell, and c(i + l, j + l), c(i – l, j— 1), c(i + l, j–l), and c(i – l, j – l) for the diagonal neighbors. A computer program starts by randomly assigning 1s and 0s to positions in the checkerboard. You can think of this as randomly painting cells white or black. One interesting simulation next examines the neighbor sites to determine if the majority of neighbors are in state one. If so, then the center site also becomes one. We can represent those cells in the on (one) state as black dots on a graphics screen. In other words, this rule is a voting rule which assigns 0 or 1 according to the “popularity” of these states in the neighborhood, and interestingly it Gargoyles generates behavior found in real physical systems. (This rule is applied to the newly generated pattern and repeated over and over again to see what patterns evolve as the 1s and 0s change.) This simple majority rule automata produces hundreds of coalesced, convex-shaped black areas but does not lead to interesting graphical forms. A way to destabilize the interface between one and zero areas is to modify the rules slightly so that a cell is on if the sum of the one sites in the Moore neighborhood is either 4, 6, 7, 8, or 9, otherwise the site is turned off. (To do this, a computer program simply examines the number of neighbor cells to see how many have a value of 1.) This rule has been studied previously. Since it uses a Moore neighborhood, it was termed M46789 by Gerard Vichniac in 1986. Such simulations have relevance to percolation and surface-tension studies of liquids.
The gargoyles on pp. 182 and 183 are M46789 forms that have evolved after a few dozen time steps from random initial conditions on a 500 × 500 square lattice. The Appendix contains BASIC and C program listings for computing numerical gargolyes.
In this chapter, we’ve had a brief introduction to gargoyles. To understand gargoyles, we must imagine the medieval person’s strong belief in God and reliance on the Church. The cathedral was a “sermon in stone” which had to be “read” by an illiterate population. Some gargoyles illustrated Bible stories, but most gargoyles were probably meant to scare away evil spirits. Religion, like the gargoyles, existed to confront and overcome chaos and danger.
Gargoyles were often of ambiguous gender and species, and perhaps these images were thought to be particularly frightening. Some have suggested that gargoyles represent pre-Christian practices and symbols incorporated into the rituals of the Catholic Church to facilitate conversions to Christianity.
In addition to these theories, I think one of the best ways to understand gargoyles is to imagine yourself being alive in the Middle Ages, a thousand years ago. Your reality is defined by the Christian Church that says that all of life, from thunderstorms to famines, is either the will of God or the jesting of the devil. Perhaps the gargoyles are a kind of spiritual test. You are placed in the universe to overcome temptations, to choose between the divine and the devil. As the centuries pass, humans become less certain of a God-controlled cosmos, and science starts offering explanations. This shift from a spirit-controlled world to scientific solutions leads to the demise of the gargoyle in the lives of people.
Today we can offer these kinds of rational analyses, but gargoyles were a mystery to many in the past. St. Bernard of Clairvaux wondered in the 12th century:
What are these fantastic monsters doing in the cloisters under the very eyes of the brothers as they read? What is the meaning of these unclean monkeys, strange savage lions and monsters? To what purpose are here placed these creatures, half beast, half man? I see several bodies with one head and several heads with one body. Here is a quadruped with a serpent’s head, there a fish with a quadruped’s head, then again an animal half horse, half goat.… Surely if we do not blush for such absurdities we should at least regret what we have spent on them.
