This year, our community is exploring Math, Stories and the Meaning of Patterns. We happen to be in the middle of exploring the mechanics of circles and their variations in geometry, number patterns, nature, and stories. We have explored Genesis 1, The Three Little Pigs, and the myths of Persephone and Perseus. I've been steeped in circles for weeks.
Then, the other day, Art of Homeschooling posted this on Instagram.
I've seen stars like this at the centers of other circles. But I had never noticed one inside an apple. Of course there would be a star at the center of an apple.
That moment of recognition left me wondering, “Why do we keep finding stars at the centers of circles?"
Circles carry stars. They are everywhere.
Just in case you think I'm being hyperbolic. . .
A Small Sampling
What are they doing there? What is happening around circles that stars keep appearing?
Notice A Circle
So let us look at a circle and see what is going on. We call that noticing and wondering.
What do you notice about a circle?
A circle is round. It has no corners. It has an inside and an outside. You can turn it, and it still looks the same. It can roll.
How is a circle formed?
Drawing A Circle
I cannot draw a perfect circle freehand. Can you? How would we draw one?
I once taught a homeschool class where we tried to draw circles using whole-arm movement. We were not really studying circles. We were studying points of rotation to create tessellations.
Turn sideways with your shoulder close to a whiteboard. Hold a marker in your outstretched hand. Quickly rotate your outstretched arm around your shoulder joint, tracing a circle on the whiteboard with the marker. Your shoulder becomes the fixed point. Your arm can move, but only as far as its length allows. Your hand traces the limit of that movement.
The point of rotation produces both the movement and its boundary.
Center and Boundary
Now we have discovered something about circles. Every point along the boundary of a circle is exactly the same distance from its center.
That is not true of a square or a triangle. Measure from the center of a square to the middle of one side, and then from the center to a corner. The distances are different. The same is true of a triangle.
But a circle has no closer points and no farther points. Its boundary is formed by one distance rotating around one fixed center. That is what makes a circle a circle.
Now we have a center and a periphery. The periphery is the boundary of the circle. What does a boundary do?
A boundary limits. It marks how far something extends and contains what's inside it.
In a circle, the radius gives us the distance from the center. The boundary is made by holding that distance as we move around the center. If the line moves closer or farther away at any point, the shape is no longer a circle.
Think about a tire with a bulge in it. It is still a tire, but its boundary no longer holds a circular form. The tire has become dangerous. The tire tells us something about what a circle does.
We count on the tire to remain round so that it rolls consistently down the road. Because every point on its boundary is the same distance from the center, the axle remains at the same height as the wheel turns.
If the tire bulges or caves in, that distance changes. The wheel may still turn, but it bumps, wobbles, and becomes dangerous. Travel depends upon the circle holding its form.
If we are good wonderers, we should ask, "What can happen inside a circle? What kinds of movement and arrangement does a circle make possible?"
Around the Boundary
Information can travel around the boundary. As a tire rolls down the road, each point along its boundary comes into contact with the ground, lifts away, travels around, and returns.
Other kinds of information can be arranged around a boundary in the same way. Spring becomes summer, summer becomes fall, fall becomes winter, and winter becomes spring again. Monday becomes Tuesday, then Wednesday, Thursday, Friday, Saturday, Sunday, and Monday again.
Where does either pattern begin? It can begin anywhere. Spring can be placed at the top of the circle, but so can winter. Sunday can begin the week, but Monday can too. The information travels around and returns. There is no final arrival. The cycle repeats.
At the Center
A circle also has a center. What is the center doing?
The center may remain still while everything turns around it. It provides the point of rotation. It gives every point along the boundary its distance and holds the entire movement in relation.
We use centers this way outside geometry. A town has a center. A home may become the center of a family's life. We orient ourselves toward a center, travel away from it, gather around it, and return to it.
So what can happen between a center and a boundary? What kinds of paths can information take between them?
Spokes
If we draw repeated paths directly from the center to the boundary, we produce spokes.
A captain's wheel gives us another arrangement. Its spokes travel from the center to the boundary, but they do not stop there. They extend beyond the circle and become handles.
Now the boundary is not simply an ending. Something can travel from the center, pass through the boundary, and continue beyond it.
The Outward Spiral
Information can also begin at the center and move outward.
Imagine wrapping a long string around a stake and tying the loose end to someone's leg. As the person walks around the stake, the string unwinds. The person continues circling the same center, but the distance between the person and the center steadily increases.
Each time the person circles back, they pass the place where they began, but they do not return to it. They are farther out.
The path they trace is a spiral. But the movement is not unlimited. The length of the string still establishes the boundary. Once the string has completely unwound, the person has reached the farthest point from the center that the string permits.
Carrying The Seasons Forward
The seasons move in a circle. Spring, summer, fall, and winter return again and again. But human beings do not encounter them as isolated cycles. We carry our previous seasons into the next.
My expectation of the coming summer is shaped by every summer I have already experienced. When winter arrives, I carry my previous winters with me. They provide both familiarity and anticipation.
If winter comes and it never freezes, never snows, and rains for months, I recognize the anomaly because this winter does not correspond to the winters I remember. The seasons cycle. Our lives move through them as a spiral. We circle back, carrying all the previous circles with us.
We call this recursion. What came before is taken up and carried into what comes next.
The Inward Spiral
Information can also move from the boundary toward the center. Think of a whirlpool. Water does not travel directly inward along a spoke. It circles as it is drawn closer and closer to the center, each turn getting progressively smaller.
The path is still a spiral, but now the movement is inward. A funnel works in much the same way. Material enters across the wide opening, circles inward, and is gathered toward a single narrow point.
Where Petals Cross
Information does not have to travel directly between the center and the boundary. It can move outward while also moving around the center.
Begin at the center. Move outward, curve around, and then return to the center. The path produces a petal.
The movement can circle in either direction. It can move outward and curve forward before returning, or move outward and curve backward before returning. In either direction, the petal is made by two movements happening together: the distance from the center changes while the position around the center also changes.
Repeat the passage at different angles, and the petals form a rose.
The petals may remain beside one another, each occupying its own part of the circle. But they can also pass through paths traced by previous petals. When those passages overlap, their intersections form a star at the center. This pattern is found in the Art of Homeschooling's apple image. But it is also found in music intervals, Rose Windows, engineering, and electromagnetism.
Multiplication in the Round
We can also arrange multiplication around a circle.
Place a set of evenly spaced points around the boundary, like the numbers on a clock. Begin at one point and draw a straight line to its multiple. Then move to the next point and do it again.
Both points move. The starting point advances one position around the circle. The ending point advances according to the multiplier. With multiplication by two, the ending point moves twice as quickly as the starting point. The lines are straight, but as they accumulate, a curve appears among them.
How clearly we can see that curve depends partly upon how many points we place around the circle. With ten or twenty points, we see a coarse arrangement of lines. Increase the number to 360, one point for every degree, and the lines become dense enough for the curve they form together to become visible.
The multiplier changes the relationship between the two points. Move the multiplier from two to three, then four or five, and the lines reorganize. One cusp becomes two, then three, then four. Move the multiplier back and forth, and one pattern opens into another.
This tells us that if we change the arrangement of information, it will make different relations available to us.
Written in the Sky
Look into the heavens, and the same geometry is written across the night sky. Earth and Venus move around the same center, the Sun, at different distances and at different rates. For every eight orbits Earth completes around the center, Venus completes roughly thirteen. If you draw a line connecting their positions every few days as they orbit, those repeating straight lines do not clutter space; they trace a dynamic, symmetrical dance. Over eight Earth years, their changing relationship sweeps out a five-petaled rose centered on the Sun. The geometric star inside the apple at your kitchen table is the same fivefold figure traced by the planets in their journey around the sun.
We are not done exploring circles or how information can be arranged in a circle. I could go on, but at some point, I have to stop.
Show Me the Patterns
Charlotte Mason called education the science of relations. Classical Christian education makes a similar claim. Reality is not divided into unrelated subjects. Truth is one, and the things we study belong to a coherent creation.
Saying that everything is connected is not the same as showing the connections. I recently finished reading a book about classical Christian education. The author said we need to teach children patterns that span subjects.
Yes. Absolutely.
But at the end of the chapter, I asked: Which patterns? Show me the patterns.
It means nothing to say that children should see the relationships among mathematics, science, literature, history, and theology if we cannot put those relationships in front of them. We cannot give children patterns that we have not learned to see.
It should not surprise us that children view the apple, the orbit, multiplication, and myth as unrelated. That is what "school as subjects" does. It puts unnecessary boundaries on creation.
The apple appears during biology or home economics. The orbit appears during astronomy. The multiplication table appears during mathematics. Myth appears during literature. Each belongs to its own subject, book, lesson, and hour. I'm not saying there aren't subjects, but those subjects naturally overlap in reality. The patterns are not decoration; they are really there.
Bringing Perseus into mathematics does not mean decorating a math lesson with a Greek myth. Perseus is not an add-on. The movement of the circle is pressing upon the story. Cycle, spiral, passage through the center, and return shape how the story moves.
The mathematics gives us another way to see what Perseus is doing. Perseus gives us another way to encounter what a circle can do.
If these things never meet, how will the child discover that they have something to say to one another?
Is Math Really Less Christian?
Someone recently told me that it is fairly easy to build a Christian curriculum, or a Charlotte Mason curriculum, in which literature, history, science, art, and theology speak to one another.
Math is harder.
When we get to math, we grab a textbook and teach whatever comes next. We argue about spiral review versus mastery, as though the way we arrange workbook pages were the only question that mattered. Math remains sealed inside its own book, as though it were somehow separate from the rest of creation.
I take offense to that.
I am not even really a math person. I am a word person. People know me because of mathematics, but I became a math person because that was my point of failure in our homeschool. I stayed because fixing that failure made me actually look.
My contention is that mathematics may be the easiest place of all to teach from a Christian point of view. Mathematics is not an isolated set of mechanical steps; it is the language of relational resonance.
If I am right, then the circle, Perseus, the sea urchin, the multiplication facts, and the polar rose are not a collection of clever connections. They are all tending in the same direction.
The circle holds unity and multiplicity together - the one and the many. It is one form containing many points, many radii, many petals, and many possible movements. The petals remain distinct without breaking the circle. The circle remains one without erasing the petals.
What does that have to say about the one and the many? What does it have to say about the Trinity, the ultimate pattern of unity without collapse and distinction without division?
What does Perseus have to say about Christ? What do rose petals have to say about the Incarnation?
If the world is patterned, and if the things Christians say about creation are true, then we should be able to look and see that the Lord is Good. We should be able to place Perseus beside the circle, the circle beside the sea urchin, the sea urchin beside multiplication, and multiplication beside the polar rose because they belong to the same creation.
They are not the same thing. But they should have something to say to one another.
And they may all have something to say about Christ.
Tender Correspondence
This is what I call Tender Correspondence.
It is not quite Charlotte Mason's science of relations. It is not merely the observation that similar patterns appear across different subjects. And it is certainly not a matter of teaching mathematics or science and then tacking Christianity onto the end.
I mean something more specific. To tender something is to offer it. A tender is something held forth as payment, something offered toward what is owed. The word also carries a sense of stretching or extending toward something else. Additionally, it means gentle, delicate, and careful.
Correspondence means resemblance. Two things correspond when we can recognize a shared form between them. But correspondence also contains response. One thing calls and another answers. They respond together.
In Tender Correspondence, the pattern in one place holds forth what it bears toward another. The circle offers something to the sea urchin. The sea urchin offers something to multiplication. Multiplication offers something to the polar rose. The rose offers something to Perseus. Perseus calls toward Christ, and Christ answers.
These are not connections we fasten onto otherwise unrelated things. The things themselves are calling back and forth. Each one stretches toward the others without becoming interchangeable with them.
The apple is not multiplication. The polar rose is not Perseus. Perseus is not Christ. They are not identical. They are not interchangeable. But they are also not unrelated.
Across number, matter, movement, story, and revelation, they are bearing witness to the same reality.
The One and The Many
A circle gives us one boundary composed of infinitely many points. Each point remains distinct, but every point is held in the same relationship to one center.
Draw the radii, and one center reaches toward many points along the boundary. Draw a spiral, and many turns belong to one continuous path. Draw a polar rose, and many petals belong to one curve. The petals remain distinct, but they meet at the center. When their paths cross, the many passages produce a star.
The one does not erase the many. The many do not break the one.
The circle is not God. A geometric pattern does not explain the Trinity. But neither is the created order utterly unrelated to its Creator. The way information is organized inside a circle bears witness to unity and distinction, the one and the many, movement outward and return toward the center.
The spoke, spiral, rose, center, boundary, and star are not identical. They cannot be substituted for one another. But they are also not unrelated. Each offers another arrangement of the same relationships and allows us to see something the others conceal.
The same is true when we place the circle beside the sea urchin, multiplication beside the polar rose, or Perseus beside Christ. They are not identical. They are not interchangeable. But they, too, are not unrelated.
They correspond.
And if these correspondences are actually present in the created order, then what are they calling toward?
If a shape can hold unity and distinction in space, a story can hold them in time. How does this same circular movement press upon stories?
Perseus: A Rose Around Death
If we arrange the story of Perseus chronologically, it forms a line. One event follows another from his birth to his death.
But what happens if we place death at the center and life at the periphery?
The story begins before Perseus is born. King Acrisius receives a prophecy that his daughter's son will kill him. He shuts Danae inside a bronze chamber so that she cannot bear a child.
Danae is enclosed in something like a tomb. But Zeus enters the chamber as a shower of gold, and life begins inside the place designed to prevent it.
Perseus is born.
Acrisius seals Danae and the infant Perseus inside a chest and throws them into the sea. Once again, Danae and Perseus are enclosed. The chest is coffin, ark, and womb. They disappear from one world and emerge alive in another.
This is the first passage. Perseus moves from life toward death and emerges into life again.
On Seriphos, Polydectes eventually sends Perseus to bring back the head of Medusa. Perseus must now approach death deliberately.
But he cannot do it through his own power.
He receives a reflective shield, winged sandals, Hades' cap, a blade, and the kibisis, a vessel capable of containing Medusa's head. Each gift allows him to approach something ordinary human capacities cannot survive.
Perseus cannot look directly at Medusa. He approaches through reflection. He conceals himself. He cuts off her head and places it inside the bag.
At the place of death, something unexpected happens. Pegasus and Chrysaor emerge from Medusa's body. Life appears from the place where death has just occurred.
Perseus passes through the center and returns carrying what happened there.
But the Medusa passage does not end when Perseus leaves her cave. He carries the head into the next part of the story.
He finds Andromeda chained to a rock, waiting to be devoured by a sea monster. Perseus enters another passage from life toward death. He kills the monster and frees her. When Andromeda's former suitor later attempts to take her, Perseus reveals Medusa's head and turns him and his followers to stone.
What Perseus gathered during one passage through death enters the next passage and changes its outcome.
Perseus then carries both Medusa's head and Andromeda home to Seriphos. There he finds that Polydectes has continued to threaten Danae. Perseus reveals the head again. Polydectes and his companions become stone. Danae is delivered, Dictys is restored, and Seriphos is released from the king's rule.
The petals are not isolated episodes. Each passage crosses the paths of the passages that came before it. Perseus carries the gifts into the encounter with Medusa. He carries Medusa into the rescue of Andromeda. He carries Medusa and Andromeda into his return to Danae.
Each cycle through death is taken up and carried into what follows.
At last, Perseus returns the gifts. Medusa's head is placed at the center of Athena's circular shield. The face that once turned the living to stone becomes protective.
Death remains death. But its place within the pattern has changed.
Perseus eventually fulfills the prophecy he was born beneath. He accidentally kills Acrisius with a discus. He leaves Argos and founds Mycenae. Perseus can approach death, survive encounters with it, and rescue others who remain under its threat. But he cannot permanently open the way out.
He passes through death and returns carrying death into life.
Then Perseus himself dies. He isn't cast into darkness. He is a hero. He is placed in the heavens as a constellation. He becomes a star.
Christ Answers Perseus
Perseus enters the territory of death while protected from it. Christ enters death itself.
Perseus cannot look directly at Medusa. He needs the shield, invisibility, flight, a sword, and a vessel to contain the head. The gifts keep death from taking him.
Christ does not evade death's gaze. He allows himself to be seen, touched, seized, condemned, wounded, and killed. He does not pass safely beside death. He gives death his own body.
That reveals the crucial difference: Perseus kills so he can live. Christ dies so that the dead can live.
Perseus brings Medusa's death-bearing head back into life, but it's wrapped safely inside a bag. Christ brings Life Himself down into the jaws of death.
Death receives Him and discovers, too late, that it has swallowed that which is impossible to contain. That is the Paschal proclamation: Christ tramples down death by death. He does not defeat death from outside its perimeter; He enters the center, and because the One entering is Life Himself, the center cannot contain Him. He emerges, not wielding death as a weapon, but despoiling it of all its power.
One of Us
We know that it is not enough for a god to solve the human problem from a safe distance. The hero must represent us. He must enter the danger as one of us, or humanity itself has not passed through.
Perseus is demigod, part human and part god. It is interesting that the only reason he can make the passage is that divine gifts temporarily give him capacities beyond ordinary humanity.
Christ is fully divine and fully human. As God, he possesses the life death cannot overcome. As man, he can truly enter human death.
Because human nature passes through death in him, his victory is not merely something done near us or presented for our observation. It is accomplished within the nature we share.
He is not merely an exceptional individual who survives. He carries humanity through. Sort of how three is carried in ten.
That is recapitulation. Christ takes up the whole human story: birth, growth, temptation, suffering, mortality, and death, and lives it toward its true end. The path remains human, but it no longer terminates in death.
The Gifts
Perseus receives many gifts so that he can cross the center and return: a shield for mediated sight, sandals for passage, a cap for concealment, a blade for action, and a bag for containing death.
Christ also receives and gives, but he is not equipped with objects that protect him from death. He himself is the gift.
The Father gives the Son. The Son gives himself. The Spirit rests upon and raises humanity. Christ gives his body and blood as life. The risen Christ gives the Spirit.
The movement in Perseus is: gift, hero survives death, hero brings back a weapon.
The movement in Christ is: Gift enters death, Gift becomes passage, Gift returns as shared life.
Perseus receives gifts that enable his victory. Christ's victory takes the form of self-gift.
Wounds that Speak: What is Carried Back
Perseus returns carrying Medusa's severed head concealed in the kibisis. Death remains dangerous. It must ordinarily remain covered because exposure to it petrifies.
Christ returns bearing his own wounds.
He does not hide them. He shows them to Thomas. They no longer pour out death. They disclose identity, love, and victory. The wounds are not erased by resurrection, but neither are they death-bearing. They have been taken into glorified life.
Perseus carries the dead other. Christ carries his own death, transformed within his living body.
Resurrection does not restore Christ to an untouched condition, as though the Crucifixion never occurred. The risen body bears the passage. However, death is not the master of those wounds. It isn't allowed to dictate the meaning.
They have become openings of recognition.
Returning for the Dead
Perseus turns his enemies to stone. His victory protects life by directing death toward those who threaten it.
Christ's victory is stranger. He destroys death without treating human beings as the final enemy. He prays for his killers. He descends among the dead not to petrify them but to lead captives out.
Perseus brings the power of death back as a weapon.
Christ returns with the keys of death and Hades because death itself has been bound. His victory is not complete when the enemy is dead. It is complete when the dead are alive.
The true hero must not merely defeat the monster. He must rescue those the monster has swallowed.
Perseus can rescue Andromeda before the monster devours her. Christ enters the belly of death for those already captive.
Perseus returns from death. Christ returns for the dead.
The Shield and the Cross
There is one more startling correspondence.
Medusa's head, once an instrument of terror, is placed at the center of Athena's circular shield and becomes protective. Here, death is literally placed in the center of the circle.
The Cross, once an instrument of imperial torture and public terror, becomes the sign of life and victory. Christianity does not forget what the Cross was. Its horror is not renamed as pleasant. But the instrument of death is taken up into Christ's self-gift and becomes the sign through which death's defeat is proclaimed.
Once again, death is not erased. Its place within the pattern changes. But in Christianity this goes beyond protection. The Cross does not merely ward death away. It has become the place through which death is entered and undone.
Perseus passes through death and returns carrying death into life. Christ passes through death and returns carrying Life into death.
Perseus is remembered in the heavens. Christ ascends into them, bearing a living body.
Perseus calls toward Christ as an unfinished pledge. Christ answers.
Taken Up Into Christ
The circle, the snail shell, the star anise, and Perseus himself are taken up into Christ. Each calls with echoes of correspondence. Each tenders what it bears.
They are not Christ. They cannot be substituted for another. But they are also not unrelated to him. They bear a resemblance and echo back and forth.
They call toward Christ because they were spoken through the Logos before they could call. And when Christ answers, he does not erase their distinctions. He gathers them into the reality toward which they were already reaching.
If we attend, we can also follow the pattern backward. From Christ to Perseus. From Perseus to the rose. From the rose to multiplication and planets. From planets to petals. From petals to the star inside the apple.
Which brings us back to the question with which we began:
Why do circles carry stars?
The patterns are knocking. Is anyone home?
