Transformed Faith: Wonder in an Arrow (Part 2 of 4)🧭

How a cracked foundation made room for color

“We thought we were the lords and masters of Mathematical Reality—not even close! We get to play there, and we certainly get to invent and discover and explore, but we most definitely do not own the place. In fact, my favorite feature of mathematics is that it continually blows my mind and teaches me humility. To be utterly and absolutely wrong is one of the greatest gifts one can receive. Luckily, mathematics is extremely generous in this regard!”

Lockhart, Paul. The Mending of Broken Bones: A Modern Guide to Classical Algebra (pp. 81-82).

A different way to see

In Part 1, the comfort of a single line began to crack—both in my math and in how I was living my faith like a ledger. More zoom on the same line didn’t help. I needed a new lens.

This series is about that kind of perspective—how a fresh view can turn something familiar into something alive. We feel that on a clear night sky, in a blazing sunset, or walking the rim of a canyon. I love those vistas. But the place I most often meet surprise and beauty is in mathematics. Math is, at its heart, the art of looking again—from another angle—and finding pattern where we didn’t expect it.

We’re used to meeting numbers on a single line: tidy, left to right. Helpful, yes—but not the only lens. What happens if we change the view? Could a different picture reveal a deeper truth about what numbers are?

Let’s spread the points out and see.

Spreading the points out (what the dots mean)

A rational number is a ratio of whole numbers. On a grid, we give that ratio two directions:

• numerator → up
• denominator → across

Start at the origin (0,0). “Up 2, across 1” stands for $\frac{2}{1}$. “Up 1, across 2” stands for $\frac{1}{2}$. Any whole “up” and whole “across” marks a rational number. With that single rule, the grid becomes a sky of fractions—every dot a ratio.

You may want to pause here. If you’re like me, you’ve spent a lifetime seeing numbers along a single line. Now they’re spread out—same numbers, new view. Let’s sit with that for a moment.

Creating an arrow from dots

Place your pencil at the origin (0,0). Now imagine a thin arrow—think a narrow beam of light—shooting from the origin toward a dot and extending it.

‘Up 2, across 1’ and the arrow pointing that way both represent $\frac{2}{1}$. Think of a narrow beam: its aim (not its length) is what carries the number.

Are we still seeing all the fractions we saw on a ruler?

Let's face it, there are a lot of fractions on a number line. Once we rearrange them in a different order, how do we know we have included all the positive fractions? It looks scattered because we stopped lining numbers up in single file. But nothing went missing. How can we be so sure?

Any fraction $\frac{b}{a}$ from the number line appears here as “up b, across a.” What changed isn’t which numbers we have—it’s how we read order.

• On the line, we ordered by distance (left → right).
• On the grid, we order by direction—how steep the arrow from the origin points.

Gentle arrows (like $\frac{1}{2}$) are “small,” the 45° arrow $\frac{1}{1}$ sits in the middle, steeper arrows (like $\frac{2}{1}$) are “larger”. Size still increases smoothly; we’ve traded distance for rotation. Once you feel that, the “scatter” becomes a clear, navigable landscape.

Order without a “next” (the paradox to notice)

Here’s the stretch: the rationals can be put in order by direction, but none of them has a next one.

Pick $\frac{3}{2}$. No matter how narrowly you squeeze the angle around it, more fractions slide in—like $\frac{10}{7}, \frac{17}{12}, \frac{24}{17}, \frac{41}{29}$ … Each one is distinct, yet a hair closer in direction. On a ruler we feel a “next tick,” but for fractions that’s an illusion: between any two comes another, and another. The grid just makes that reality visible—order without successors.

That tension—clearly ordered, yet endlessly refinable—is part of the wonder we’re walking into.

Lines of sameness (equivalence you can see)

Aim one arrow through “up 1, across 1.” It also passes “up 2, across 2,” “up 3, across 3,” … different dots, same direction, same value. Aim through “up 1, across 2,” and you’ll hit “up 2, across 4,” “up 3, across 6,” … again same value. Reducing a fraction is just “find the first dot on that arrow,” the one closest to the origin. Equivalence stops being a rule to memorize and becomes a picture you can see.

Notice we can identify the next dot on the arrow simply by repeating the pattern from the previous dot. For example, from the origin, we can choose the pattern up 1 across 1 and continue that pattern from each dot on the gray line to hit the next dot on this line.

Likewise if the pattern is up 1 across 2, then if we continue the pattern "up 1, across 2", we hit all the dots on the red line.

Now that we have dots spread out on a grid with space to breathe, we can imagine other patterns that may produce something interesting.

A playful pattern with a little zip

We can experience a different type of equivalence by "walking on the grid" repeating a simple move. Start at the point up 1, across 1 (that’s $\frac{1}{1}$). Now repeat this two-step rule:

1. Across step: go across by the previous up amount.
2. Up step: from there, go up by $2 \times$ the previous across amount.

Run it a few times:

• Start at (up 1, across 1). Then begin the pattern from this point:
  Across by previous up (1) → across becomes 2;
  Up by $2 \times$ previous across $2 \times 1 = 2$ → up becomes 3. Together (up 3, across 2) → $\frac{3}{2}$.
• From (3,2): across by previous up (3) → across 5;
  up by $2 \times$ previous across $2 \times 2=4$ → up 7. Together (7,5) → $\frac{7}{5}$.
• From (7,5): across (+7 → 12); up (+ ($2 \times 5=10$) → 17) → $\frac{17}{12}$ .

As you plot these points, they seem to line up—almost as if one arrow could pass through all of them. But look closely: they’re not exactly on a single straight line.

Each point determines a slightly different arrow, all pointing in nearly the same direction. Remember, these were the same fractions we viewed in Part 1 on a number line. But now, in two-dimensions, we view the pattern that generated these fractions.

And this pattern points us to the question that changes everything:

The question that changes everything

Once you can see the rationals and order them by arrows, you can finally ask:

On the number line, that question can’t breathe; points blur together. In the plane, each rational is a distinct dot, so “miss every dot” becomes meaningful. If such an arrow exists, what number is it pointing to? Is it just a long decimal—or a different kind of number?

Feel the scale (the “sun” moment)

The sun is about 93,000,000 miles away. Hold that number.

Now picture three arrows from the origin. Two of them are rational arrows that don’t hit any lattice point until they finally land way out at heights beyond 93 million. They skim past dot after dot after dot… and only then clip a lattice point for the very first time.

In other words, along that direction the “next dot on the grid” is over 93 million steps away.

Between those two arrows slips a third yellow arrow—flatter than the top arrow, steeper than the bottom arrow—that threads the sky forever and never hits a single dot. Before that 93-million mark, after it, always. It runs the heavens without colliding once.

That’s not “more digits.” That’s a different destiny. It is what we call irrational.

Naming the arrow

It may seem impossible to engineer such an arrow. However, there’s a simple way to draw one: take a unit square. Its diagonal has length $\sqrt{2}$. Point an arrow with that slope. On our grid, it lives between the rational arrows through (up 3, across 2) and (up 7, across 5) etc., always threading the gap, never landing. The rationals paint the stars; this arrow is the thin, exact line of color flowing between them.

This graph shows a bigger picture of the near hits.

Even though it appears the line intersects the points, it is clear it does not if you zoom in to enough detail as the next graph illustrates.

This beautiful perspective of an infinite arrow missing every dot also impacted my faith.

What this did to my faith

From dots to Light

Up close, the arrow never quite touches a dot. That picture started reading me.
For years I lived like the number line—always reaching for the next tick, the next proof I was right. The dots were my comfort: tidy, checkable, earned. But the grid taught me a truer thing about faith. The dots are good—like commands that mark the sky—yet the Light that threads between them is what gives the sky its meaning. Jesus doesn’t erase the dots; he fulfills and illumines them, showing the mercy they were meant to carry.

Cracks—and a new posture

I began to see my cracks: I chased the “right dot” and missed the grace between the dots. I measured well and loved thin. The arrow became a parable—near every point, touching none—a picture of holiness that fulfills the law without contradiction and of grace that draws close without crushing. The dots still guide my steps; the Light gives my direction. Obedience stopped being self-vindication and became love in motion—truth with mercy, justice with humility.

Not irrational—super-rational

I picture that needle-thin, endless beam as Jesus—the Light of the world. In math, a direction that never hits a dot is tied to what we call an “irrational” slope; outside math, that word can mislead.
So to be clear: I’m not calling faith irrational. I think of it as super-rational—beyond what I can fully compute, yet deeply coherent to my analytical mind.

Living in color

This picture has transformed how I think about numbers—and about following that infinite beam. It feels like moving from managing lines to navigating by Light: from black-and-white control to color that keeps its shape.
In color, the differences aren’t problems to fix; they’re facets to notice—angles of the same heart-ward direction. I still try to honor the dots that give form, but I’m learning to delight in the hues between them: the varied stories, tempos, and gifts of people made to reflect a larger Beauty.

Where we go next

We’ve seen that a “miss-every-dot” arrow exists—and felt its scale. But “$\sqrt{2}$ is irrational” can still sound like a line from a textbook: true, yet distant. How do we know it, not just hear it? Can we move from being told a fact to seeing why it must be true? Better yet—could the story of why be even more beautiful than the fact that it is?

In Part 3, we answer with a picture. One image. Three squares. No heavy algebra—just a quiet jigsaw that turns wonder into certainty. The irrational arrow you met here becomes inevitable there: a proof that paints in color.

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