Module 1 · Lesson 2

Geometry: Curved space, GPS, and the packing of spheres

You'll see geometries in which triangles' angles add up to more or less than 180 degrees, and why your phone's GPS has to account for the curvature of space and time. You'll also meet a centuries-old question about packing spheres, solved in three dimensions and still open in most others.

What you will be able to do

Describe how non-Euclidean geometries differ from Euclid's and where geometry is used today, and state one open problem.

0% of this lesson · 10 items · 1h 33m total · 1h 18m without the optional activity

Contents of this lesson10 items

ReadingYour Phone's Location Depends on Curved Spacetime3 min
ReadingSpherical and Hyperbolic Geometry: Triangles That Don't Add Up to 1804 min
ReadingHow GPS Finds You: Distances, Clocks, and Relativity4 min
ReadingPacking Spheres: Kepler's Oranges and Viazovska's Eight Dimensions4 min
ReadingAn Open Problem: Kissing Numbers in Higher Dimensions4 min
Guided ReadingGuided Walkthrough: Locating a Receiver from Three Distances7 min
Guided ConversationIs Your World Flat or Curved?12 min
Hands-on Activity · optionalDraw a Triangle with Three Right Angles15 min
Knowledge CheckCurved space, GPS, and the packing of spheres10 min
Graded QuizGeometry30 min

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Reading 3 min Done

Your Phone's Location Depends on Curved Spacetime

Open a map on your phone and a dot shows where you are, often to within a few meters. It's easy to assume the phone works this out by itself. It doesn't. It listens to radio signals from satellites orbiting about 20,000 kilometers overhead, and it works out its position from how long those signals took to arrive (Ashby 2002).

That method depends on time, measured very precisely. Each satellite carries an atomic clock, and each signal says when it was sent. Radio signals travel at the speed of light, about 300,000 kilometers per second. So an error of one millionth of a second in a satellite's clock becomes an error of about 300 meters in the distance your phone computes.

Here is the puzzle. A clock that keeps perfect time on the ground doesn't keep the same time once it's in orbit. It runs fast. According to the physicist Neil Ashby, the net effect is that a satellite clock gains about 4.46 parts in ten billion relative to a ground clock (Ashby 2002). That sounds negligible. Over one day it adds up to about 38 microseconds, or 38 millionths of a second.

Multiply 38 microseconds by 300 meters per microsecond and you get more than 11 kilometers. Ashby puts it directly: left uncorrected, the clock errors that build up in just one day would cause navigation errors of more than 11 kilometers (Ashby 2002). Your map dot would drift across town by evening and keep going.

Why would two well-made clocks disagree? The answer comes from Einstein's theories of relativity. Two effects pull in opposite directions:

A clock higher up in Earth's gravity runs faster than one on the ground. The satellites are far above you, so their clocks gain time.
A moving clock runs slower than one at rest. The satellites travel at about 4 kilometers per second, so their clocks lose a little time.

The first effect is larger, so the satellite clocks run fast overall (Ashby 2002). GPS engineers build the correction into the system so that your phone gets the right answer.

The mathematics behind those effects is geometry, but not the geometry of a flat sheet of paper. General relativity describes gravity as the curving of space and time together. In that geometry, the rules you may remember from school, such as the angles of a triangle adding up to 180 degrees, hold only approximately, over small regions. The idea that geometry could work any other way began as a question about one of Euclid's assumptions. It now runs, unnoticed, inside every phone that shows you where you are.

This content reflects the field as of October 2026.

References

Ashby, Neil. 2002. "Relativity and the Global Positioning System." Physics Today 55 (5). doi:10.1063/1.1485583.
- Free: None confirmed
- Publisher: Physics Today
- Free companion: Ashby 2003

Reading 4 min Done

Spherical and Hyperbolic Geometry: Triangles That Don't Add Up to 180

Introduction

In Euclid's geometry, the angles of every triangle add up to 180 degrees. That result depends on Euclid's assumption about parallel lines, and when you change the assumption, triangles change too. This reading describes two geometries that differ from Euclid's, one where triangles' angles add up to more than 180 degrees and one where they add up to less, and what they mean for the geometry of real space.

Geometry on a Sphere

On the surface of a ball, there are no straight lines in the usual sense. The role of a straight line is played by a great circle: a circle on a sphere whose center is the center of the sphere, such as the equator. The shortest path along the surface between two points always follows a great circle. That is why a long flight, drawn on a flat map, looks like a curve.

Spherical geometry is the geometry of the surface of a sphere, with great circles as its lines. Any two great circles cross, in fact at two opposite points. So on a sphere, there are no parallel lines at all.

Triangles behave differently, too. Start at the North Pole, travel straight down a line of longitude to the equator, turn and travel a quarter of the way around the equator, then turn and travel straight back up to the North Pole. Each of the three corners is a right angle. The triangle's angles add up to $90 + 90 + 90 = 270$ degrees.

Figure: A globe with a triangle drawn on its surface. One corner is at the North Pole; the other two are on the equator, a quarter of the way around from each other. Each side follows a great circle, and each of the three corners is marked as a right angle. Alt text: A sphere with a triangle whose corners are the North Pole and two equator points a quarter-turn apart, showing three right angles.

On a sphere, every triangle has an angle sum greater than 180 degrees. The larger the triangle, the larger the sum. A small triangle, such as one drawn on a town-sized patch of the Earth, has a sum very close to 180, which is why flat geometry works well for a building or a town.

Geometry with Many Parallels

The second alternative goes the other way. Hyperbolic geometry is a geometry in which, through a point not on a given line, there are many lines that never meet the given line. In it, triangles' angles add up to less than 180 degrees. As early as 1766, Johann Lambert noticed that in this kind of geometry, a triangle's angle sum grows as its area shrinks (O'Connor and Robertson 1996a). Large triangles have small angle sums; tiny ones come close to 180.

In 1868, Eugenio Beltrami showed that this geometry could be modeled on a curved surface called a pseudosphere, shaped somewhat like the bell of a trumpet (O'Connor and Robertson 1996a). The table below compares the three geometries.

Geometry	Lines through a point that never meet a given line	Triangle angle sum	How the space curves
Euclidean	Exactly one	Exactly 180 degrees	Flat
Spherical	None	More than 180 degrees	Curves like a ball
Hyperbolic	Many	Less than 180 degrees	Curves like a saddle

Riemann's Wider View

In a lecture given on June 10, 1854, and published only in 1868, Bernhard Riemann treated geometry far more generally. A geometry, in his view, is any space with enough extra structure to measure things like length (O'Connor and Robertson 1996a). The space could be flat or curved, and the curving could change from place to place.

Riemann introduced a precise way to measure curvature: how far a space departs from being flat at a given point and in a given direction (Riemann [1854] 1873, pt. II, sec. 2). A sphere has positive curvature, a hyperbolic space negative curvature, and Euclid's plane zero.

Which Geometry Fits Physical Space?

Once more than one geometry is logically consistent, logic can't decide which one describes the space we live in. Riemann said so directly: the properties that distinguish physical space from other possible spaces are "only to be deduced from experience" (Riemann [1854] 1873, Plan of the Investigation). That makes it a question for measurement. General relativity later answered part of it: near massive bodies, the geometry of space and time is curved, by amounts that precise clocks can detect.

Conclusion

Spherical and hyperbolic geometry are consistent alternatives to Euclid's, with angle sums above and below 180 degrees. Riemann's framework includes all three as spaces of different curvature. Which geometry fits physical space is settled by measurement, region by region, not by proof.

This content reflects the field as of October 2026.

Key Terms

Great circle: A circle on a sphere whose center is the center of the sphere; it plays the role of a straight line, and the shortest path between two points on the surface follows one.
Spherical geometry: The geometry of the surface of a sphere, with great circles as lines; it has no parallel lines, and triangles' angles add up to more than 180 degrees.
Hyperbolic geometry: A geometry in which many lines through a point never meet a given line, and triangles' angles add up to less than 180 degrees.
Curvature: A measure of how far a space departs from being flat at a given point and in a given direction.

References

O'Connor, J. J., and E. F. Robertson. 1996a. "Non-Euclidean Geometry." MacTutor History of Mathematics Archive, University of St Andrews. Last updated February 1996.
- Free: MacTutor: Non-Euclidean geometry
Riemann, Bernhard. (1854) 1873. "On the Hypotheses Which Lie at the Bases of Geometry." Translated by William Kingdon Clifford. Nature 8: 14–17, 36–37.
- Free: Clifford's translation, edited by D. R. Wilkins (Trinity College Dublin)

Reading 4 min Done

How GPS Finds You: Distances, Clocks, and Relativity

Introduction

A GPS receiver, such as the one in your phone, can find its position anywhere on Earth without sending a single signal. It does this with geometry and with clocks, and the clocks have to be corrected using relativity. This reading explains how a receiver turns timing into distances, distances into a position, and why the satellites' clocks need correcting.

Distances from Timing

Each GPS satellite carries a satellite clock: an atomic clock on board that sets the timing of the satellite's signals. The satellite continually broadcasts its own position and the time on its clock. A receiver notes when each signal arrives. The delay between sending and arriving, multiplied by the speed of light, gives the distance to the satellite (Ashby 2002).

Light is fast, so timing has to be extremely precise. Light covers about 300 meters in a microsecond, a millionth of a second. Ashby notes that a timing error of one nanosecond, a billionth of a second, leads to a positioning error of about 30 centimeters (Ashby 2003). A clock error of one microsecond would put the distance off by about 300 meters.

From Distances to a Position

Knowing your distance from one point tells you only that you're somewhere on a circle around it. Knowing your distances from several known points narrows that down. Trilateration is finding a position from its distances to points whose positions are known. On a flat map, three distances to three points not in a line pin down a single location: two circles generally cross at two points, and the third circle picks one of them.

Real GPS works in three dimensions, where each distance puts you somewhere on a sphere around a satellite. There's a further complication. A phone's clock is far less accurate than a satellite's, so the receiver doesn't know the exact time when a signal arrives. It treats its own clock error as one more unknown. That makes four unknowns, three for position and one for time, so the receiver uses signals from four satellites (Ashby 2003).

Clocks That Tick at Different Rates

Relativity says that two accurate clocks in different situations won't keep the same time. Two effects matter for GPS.

The first is the gravitational frequency shift: a clock higher up in a gravitational field runs faster than an identical clock lower down. GPS satellites orbit about 20,000 kilometers above Earth's surface, where gravity is weaker, so this effect makes their clocks run fast compared with clocks on the ground (Ashby 2002).

The second is time dilation: a moving clock runs slower than an identical clock at rest. GPS satellites travel at about 4 kilometers per second, so this effect makes their clocks run a little slow (Ashby 2002).

For GPS satellites, the gravitational effect is larger. The net result, in Ashby's figures, is that the satellite clocks run fast by about 4.46 parts in ten billion (Ashby 2002). There are 86,400 seconds in a day, so this works out to about 38 microseconds gained each day.

Building In the Correction

At about 300 meters per microsecond, 38 microseconds a day is more than 11 kilometers. Ashby states that the clock errors from a single day would produce navigation errors of more than 11 kilometers if nothing were done (Ashby 2002). So, before launch, the satellite clocks are deliberately adjusted to tick at a slightly lower frequency, chosen so that once they're in orbit they keep pace with clocks on the ground (Ashby 2002; Ashby 2003). Further, smaller corrections are applied as the satellites move along their slightly oval orbits.

None of this is visible when you use a map. The receiver solves the geometry of spheres and the arithmetic of clocks, using rates corrected for relativity, many times a minute.

Conclusion

A GPS receiver finds its position by trilateration, from timed signals sent by at least four satellites. The satellite clocks run fast by about 38 microseconds a day, because the gravitational effect outweighs time dilation. Engineers correct for this, and without the correction, positions would drift by more than 11 kilometers a day.

This content reflects the field as of October 2026.

Key Terms

Satellite clock: An atomic clock on board a GPS satellite that sets the timing of the satellite's signals.
Trilateration: Finding a position from its distances to points whose positions are known.
Gravitational frequency shift: The effect by which a clock higher up in a gravitational field runs faster than an identical clock lower down.
Time dilation: The effect by which a moving clock runs slower than an identical clock at rest.

References

Ashby, Neil. 2002. "Relativity and the Global Positioning System." Physics Today 55 (5). doi:10.1063/1.1485583.
- Free: None confirmed
- Publisher: Physics Today
- Free companion: Ashby 2003
Ashby, Neil. 2003. "Relativity in the Global Positioning System." Living Reviews in Relativity 6: 1. doi:10.12942/lrr-2003-1.
- Free: PubMed Central

Reading 4 min Done

Packing Spheres: Kepler's Oranges and Viazovska's Eight Dimensions

Introduction

Grocers stack oranges in pyramids, each layer nestled in the hollows of the layer below. Is that the tightest way to pack them? The question took almost four centuries to answer in three dimensions, and it has been answered in only a few others. This reading explains the problem, the proofs in dimensions 3, 8, and 24, and why anyone cares about dimensions beyond three.

Kepler's Oranges

Sphere packing is the problem of arranging equal balls so that they don't overlap and fill as much of space as possible. The measure of success is density: the fraction of space the balls fill. The grocer's arrangement fills $\pi/\sqrt{18}$ of space, about 74 percent (Hales et al. 2017).

In 1611, in a short booklet about snowflakes, Johannes Kepler claimed that no packing of equal balls is denser than this one (Hales et al. 2017). That claim became known as the Kepler conjecture: the statement that no packing of equal spheres in three-dimensional space has a density greater than the grocer's stack, about 74 percent. It's easy to believe and very hard to prove, because a proof has to rule out every possible arrangement, including irregular ones no one would think to try.

A Proof Checked by Machine

Samuel Ferguson and Thomas Hales established the conjecture in 1998, but their proof wasn't published in full until 2006 (Hales et al. 2017). It relied on a large amount of computer calculation, and the referees, after years of work, couldn't certify every part of it. The proof was published without their complete certification.

Hales then led a project to remove the doubt. He and 21 coauthors produced a formal proof: one written in a precise language that computer programs called proof assistants can check, step by step, down to the basic rules of logic. They used two such programs, HOL Light and Isabelle, and published the completed proof in 2017 (Hales et al. 2017).

Beyond Three Dimensions

A dimension, here, is the number of coordinates needed to give the position of a point: two on a flat map, three in space. Nothing stops mathematicians from working with points that need 8 or 24 coordinates. Distances there are computed the same way as in the plane, by the Pythagorean theorem extended to more coordinates, so packing balls in those spaces is a well-defined problem.

Some of the best packings in higher dimensions are built on a lattice: a regular arrangement of points that repeats in every direction, like a grid, so that it looks the same from every point. Put a ball at each point and you have a packing.

In 2016, Maryna Viazovska proved that a lattice called E8 gives the densest packing of identical spheres in 8 dimensions. Shortly after, working with Henry Cohn, Abhinav Kumar, Stephen Miller, and Danylo Radchenko, she proved that the Leech lattice gives the densest packing in 24 dimensions. Her method drew on the theory of modular forms, a field that had seemed far from packing problems. She received a Fields Medal in 2022, cited for the 8-dimensional proof and for related work (International Mathematical Union 2022).

With these proofs, the problem had been solved in only five dimensions: 1, 2, 3, 8, and 24 (Cohn 2017). In every other dimension, the densest packing is unknown.

Why Higher Dimensions Matter

The higher-dimensional problem has a practical side. As Cohn puts it, "sphere packings are error-correcting codes for a continuous communication channel" (Cohn 2017). A signal can be described by a long list of numbers, which is a point in a space of many dimensions. Noise nudges the point slightly. If the allowed signals sit at the centers of non-overlapping balls, a slightly nudged signal still lies in the right ball, and the receiver can tell which one was sent. Denser packings let more distinct signals share the same space. Real communication channels can be modeled with high-dimensional spaces, which is why these packings have practical importance (Cohn 2017).

Conclusion

The densest packing of equal spheres is known in dimensions 1, 2, 3, 8, and 24. Hales's three-dimensional proof is now checked by machine, and Viazovska's 8-dimensional proof was cited in her Fields Medal. In all other dimensions, the question remains open.

This content reflects the field as of October 2026.

Key Terms

Sphere packing: The problem of arranging equal balls so that they don't overlap and fill as much of space as possible.
Density: The fraction of space that a packing of balls fills.
Kepler conjecture: The statement that no packing of equal spheres in three-dimensional space has a density greater than the grocer's stack, about 74 percent.
Dimension: The number of coordinates needed to give the position of a point.
Lattice: A regular arrangement of points that repeats in every direction, like a grid, so that it looks the same from every point.

References

Cohn, Henry. 2017. "A Conceptual Breakthrough in Sphere Packing." Notices of the American Mathematical Society 64 (2). doi:10.1090/noti1474. Preprint arXiv:1611.01685.
- Free: arXiv 1611.01685
Hales, Thomas, Mark Adams, Gertrud Bauer, Dat Tat Dang, and 18 others. 2017. "A Formal Proof of the Kepler Conjecture." Forum of Mathematics, Pi 5: e2. Preprint arXiv:1501.02155.
- Free: arXiv 1501.02155
International Mathematical Union. 2022. "Maryna Viazovska." Fields Medal citation.
- Free: International Mathematical Union (PDF)

Reading 4 min Done

An Open Problem: Kissing Numbers in Higher Dimensions

Introduction

Put a coin flat on a table and surround it with identical coins, each touching it. You can fit exactly six. The same question in higher dimensions is a long-standing open problem in geometry, and in 2025 it became a contest between human mathematicians and an AI system. This reading explains the problem, what is known, and where the records stand.

Coins Around a Coin

The kissing number of a dimension is the largest number of equal spheres that can touch one central sphere of the same size without overlapping one another (Boyvalenkov, Dodunekov, and Musin 2012). In two dimensions the spheres are circles, like the coins, and the answer is 6. In three dimensions, with balls around a ball, the answer is 12 (Aalto University 2025).

The two-dimensional answer can be checked with geometry you already know. When two equal coins touch the central coin and each other, the three centers form a triangle with all sides equal, two radii long. Each angle of that triangle is 60 degrees, so each outer coin takes up 60 degrees of the full turn around the center. Six of them use exactly 360 degrees, with no room left for a seventh.

Three dimensions already caused trouble. In 1694, Isaac Newton and David Gregory disagreed about the answer. Newton held that it was 12, and he turned out to be right, but a detailed proof came only in 1953 (Boyvalenkov, Dodunekov, and Musin 2012).

Bounds Instead of Answers

The exact kissing number is known in only six dimensions: 1, 2, 3, 4, 8, and 24. The answers are 2, 6, 12, 24, 240, and 196,560. The four-dimensional case was settled in 2003 by Oleg Musin; the cases of 8 and 24 dimensions were settled in 1979 (Boyvalenkov, Dodunekov, and Musin 2012).

In every other dimension, mathematicians know only a range. A lower bound is a number the answer is proved to reach or exceed, usually by exhibiting an arrangement with that many spheres. An upper bound is a number the answer is proved not to exceed, usually by an argument that no arrangement can do better. Where the two bounds differ, the kissing number is an open problem: a question whose answer is not yet known, though people are working on it.

A Race Between People and Machines

Before 2025, the bounds for dimensions below 16 hadn't moved in about two decades (Aalto University 2025). Then, in May 2025, Google DeepMind's AlphaEvolve system raised the lower bound in dimension 11 to 593. That was a case of AI-assisted search: using an AI system to search a vast space of possible constructions for one that beats a known record.

Later in 2025, Mikhail Ganzhinov, a doctoral researcher at Aalto University in Finland, established three new lower bounds: 510 in dimension 10, 592 in dimension 11, and 1,932 in dimension 14. In dimensions 10 and 14 these beat the previous records; in dimension 11 he came one short of the AI's 593 (Aalto University 2025). His method narrowed the search by looking only at arrangements with a high degree of symmetry. The same news release reported new results by Henry Cohn and Anqi Li in dimensions 17 to 21, the first progress there in over 50 years (Aalto University 2025).

The pattern is new. AI systems and human mathematicians now trade records on the same open problem, sometimes within months.

Why It's So Hard

Each new dimension adds another direction for spheres to point in. The number of ways to arrange them grows far faster than any search can keep up with. A lower bound needs only one good arrangement, but finding it means searching an enormous space. An upper bound needs a proof that no arrangement anywhere does better. In most dimensions the gap between the two bounds is still wide.

Conclusion

The kissing number is known exactly in only six dimensions, and in most others only lower and upper bounds are known. Records changed several times in 2025, from both an AI system and human researchers. The problem remains open in most dimensions.

This content reflects the field as of October 2026.

Key Terms

Kissing number: The largest number of equal spheres that can touch one central sphere of the same size without overlapping one another.
Lower bound: A number the answer is proved to reach or exceed, usually by exhibiting an arrangement with that many spheres.
Upper bound: A number the answer is proved not to exceed, usually by an argument that no arrangement can do better.
Open problem: A question whose answer is not yet known, though people are working on it.
AI-assisted search: Using an AI system to search a vast space of possible constructions for one that beats a known record.

References

Aalto University. 2025. "Researcher Cracks New 'Kissing Number' Bounds, Besting AI in the Process." News release, October 23, 2025.
- Free: Aalto University
Boyvalenkov, Peter, Stefan Dodunekov, and Oleg R. Musin. 2012. "A Survey on the Kissing Numbers." Serdica Mathematical Journal 38 (4): 507–22. Preprint arXiv:1507.03631.
- Free: arXiv 1507.03631

Guided Reading 7 min Done

Guided Walkthrough: Locating a Receiver from Three Distances

Introduction

A GPS receiver finds where it is from nothing but distances to points whose positions it knows. The geometry behind that is the Pythagorean theorem and the equation of a circle. This reading walks through a flat, two-dimensional version of the problem, finds the receiver's position, and then shows what a clock error of one millionth of a second does to the answer.

The Starting Point

Picture a flat map with a grid on it, measured in kilometers. Each point on the map has two coordinates: how far east of a starting corner it is, written first, and how far north, written second. Three beacons sit at known positions:

Beacon	Position (km)
A	(0, 0)
B	(10, 0)
C	(0, 10)

A receiver somewhere on the map has timed a signal from each beacon and turned each delay into a distance. Signals travel at the speed of light, about 300 meters per microsecond (Ashby 2003). The distances are:

5 km to A
$\sqrt{65}$ km, about 8.06 km, to B
$\sqrt{45}$ km, about 6.71 km, to C

The question is where the receiver is.

One tool does all the work. The distance between two points on the grid comes from the Pythagorean theorem: the east-west gap and the north-south gap form the two shorter sides of a right triangle, and the distance is the longest side. So a point $(x, y)$ is at distance $r$ from a point $(p, q)$ exactly when

(x - p)^2 + (y - q)^2 = r^2

The set of all points at distance $r$ from $(p, q)$ is a circle of radius $r$ around it, so this is the equation of that circle.

Walking Through the Location Fix

Step 1: Draw a circle around beacon A

The receiver is 5 km from A. That doesn't say which direction, so the receiver could be anywhere on a circle of radius 5 around A. With A at $(0, 0)$ , the circle's equation is

x^2 + y^2 = 25

One distance has narrowed the position from anywhere on the map to somewhere on one circle.

Step 2: Add the circle around beacon B

The receiver is also $\sqrt{65}$ km from B, at $(10, 0)$ . That puts it on a second circle:

(x - 10)^2 + y^2 = 65

The receiver must lie on both circles, so it is at a point where they cross. To find the crossing points, subtract the first equation from the second. The $y^2$ terms cancel, and multiplying out $(x - 10)^2 = x^2 - 20x + 100$ leaves

-20x + 100 = 40

So $x = 3$ . Putting $x = 3$ into $x^2 + y^2 = 25$ gives $y^2 = 16$ , so $y$ is 4 or $-4$ . Two candidate points remain: $(3, 4)$ and $(3, -4)$ . Two circles usually cross at two points, and two distances alone can't tell which point is right.

Step 3: Add the circle around beacon C

The third distance settles it. The receiver is $\sqrt{45}$ km, about 6.71 km, from C at $(0, 10)$ . Check each candidate:

From $(3, 4)$ to C: the gaps are 3 and 6, so the distance squared is $9 + 36 = 45$ . That matches.
From $(3, -4)$ to C: the gaps are 3 and 14, so the distance squared is $9 + 196 = 205$ . The distance is about 14.3 km, which doesn't match.

Only $(3, 4)$ fits all three distances. That's the receiver's position. This method, finding a position from distances to points whose positions are known, is called trilateration.

Step 4: Check each distance with the Pythagorean theorem

A good habit is to check the answer against every piece of information you started with:

To A at $(0, 0)$ : $3^2 + 4^2 = 9 + 16 = 25$ , and $\sqrt{25} = 5$ . Correct.
To B at $(10, 0)$ : $(3 - 10)^2 + 4^2 = 49 + 16 = 65$ . Correct.
To C at $(0, 10)$ : $3^2 + (4 - 10)^2 = 9 + 36 = 45$ . Correct.

All three distances check out, so $(3, 4)$ is the one point that fits.

Step 5: Add a clock error of one microsecond

Now suppose beacon C's clock is off by one microsecond. The receiver turns time into distance at about 300 meters per microsecond, so it now thinks C is about 0.3 km farther away than it really is: about 7.01 km instead of 6.71 km.

The circles around A and B haven't changed, and they still cross at $(3, 4)$ . But the circle around C has grown by 0.3 km, and it no longer passes through $(3, 4)$ . The circles around A and C now cross near $(3.26, 3.79)$ , about a third of a kilometer from the true position. The three circles no longer meet at one point. The receiver can see that something is inconsistent, but it can't tell from the circles alone which beacon is wrong.

Step 6: Scale up to a day of uncorrected drift

GPS satellite clocks, left uncorrected, would run fast by about 38 microseconds a day because of the effects of relativity (Ashby 2002). At 300 meters per microsecond, that's

38 \times 300 = 11{,}400 \text{ meters}

or about 11.4 km of error in each distance after one day. Ashby states that one day's clock errors would cause navigation errors of more than 11 km (Ashby 2002). A one-microsecond error moved the position by a third of a kilometer. A day's drift would make the method useless within hours.

Key Considerations

Three dimensions, four satellites. The flat map simplifies the real problem. In space, each distance puts the receiver on a sphere around a satellite, not a circle. The receiver's own clock is also far less accurate than the satellites' atomic clocks, so it treats its own clock error as a fourth unknown alongside its three coordinates. Four unknowns need four distances, which is why GPS receivers use signals from at least four satellites (Ashby 2003).

Distance comes from time. Every distance in this walkthrough came from a timed signal. That's why the clock error in Step 5 matters so much: a tiny error in time becomes a large error in distance.

The common mistake: thinking GPS sends your location to the satellites. It's natural to assume the satellites track your phone. They don't. Each satellite broadcasts its own position and the time, and the receiver only listens. All the geometry in Steps 1 to 4 happens inside the receiver. The satellites never learn where you are.

Summary

Three distances to three known points pin down one position on a flat map, and the Pythagorean theorem both finds it and checks it. Here is the model result:

Beacon Position (km) Distance to receiver (km) Check at (3, 4)
A (0, 0) 5 $3^2 + 4^2 = 25$
B (10, 0) $\sqrt{65}$ , about 8.06 $7^2 + 4^2 = 65$
C (0, 10) $\sqrt{45}$ , about 6.71 $3^2 + 6^2 = 45$
Position found: (3, 4).
Effect of a 1-microsecond error in C's clock: the distance to C reads about 7.01 km, the three circles no longer meet at one point, and the circles around A and C cross about 0.33 km from the true position. A 38-microsecond daily drift would add about 11.4 km of error.

Beacon	Position (km)	Distance to receiver (km)	Check at (3, 4)
A	(0, 0)	5	$3^2 + 4^2 = 25$
B	(10, 0)	$\sqrt{65}$ , about 8.06	$7^2 + 4^2 = 65$
C	(0, 10)	$\sqrt{45}$ , about 6.71	$3^2 + 6^2 = 45$

The circles around A and B cross at two points, $(3, 4)$ and $(3, -4)$ .
The circle around C rules out $(3, -4)$ , which is about 14.3 km from C.
The check uses the same Pythagorean distance formula that set up the circles.

References

Ashby, Neil. 2002. "Relativity and the Global Positioning System." Physics Today 55 (5). doi:10.1063/1.1485583.
- Free: None confirmed
- Publisher: Physics Today
- Free companion: Ashby 2003
Ashby, Neil. 2003. "Relativity in the Global Positioning System." Living Reviews in Relativity 6: 1. doi:10.12942/lrr-2003-1.
- Free: PubMed Central

Guided Conversation 12 min Done

Is Your World Flat or Curved?

In this conversation you'll test flat and curved geometry against places and trips from your own life, from a room to a long flight. You'll leave with a short, reasoned answer about which geometry suits your town and which suits the whole Earth.

You'll have this conversation with an AI assistant, using your own account. Choose a button to open a new chat with the prompt already filled in, then press send to start. If the chat opens empty, copy the prompt and paste it in.

Run this conversation in whichever assistant you already use:

Open in Claude Open in ChatGPT

Claude desktop app

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Show the full prompt (it lists misreadings to watch for, so skip it if you would rather come to the conversation fresh)

Guided Conversation: Is Your World Flat or Curved? (about 12 minutes)

Note to the learner: press send to start. Everything below is facilitator guidance for the AI. It lists misconceptions to watch for, so skip it if you'd rather come to the conversation fresh.

Please facilitate a reflective dialogue with me. I'm an adult who is comfortable with basic high-school algebra but may be rusty or uneasy with math, exploring what geometry is and whether to study it further. I'm studying geometries that differ from Euclid's and where geometry is used today. Follow this guidance for the whole conversation.

GOAL
Help me describe how non-Euclidean geometries differ from Euclid's flat geometry, and where geometry is used today, by drawing on places, maps, and trips from my own experience.

HOW TO RUN THE CONVERSATION
- Ask one question at a time, then wait for my reply. Keep each of your turns under about 120 words.
- Don't lecture. Explain a point only when I need it to continue, then return to my examples.
- Be curious, collegial, and patient. Define terms briefly on first use, such as great circle (a circle on a sphere whose center is the sphere's center, like the equator). Welcome disagreement when I give a reason.
- Write any numbers in plain text. No equations.
- Plain conversation only: don't search the web or create files or documents.
- Aim for about 12 minutes, with most of the time on topics 1 and 2. If my replies are brief, offer one concrete prompt, such as "Think of a flight you've taken or would like to take between two continents," and move on. If I seem uncertain, shorten the conversation to 5-7 minutes. Always reach the final topic.
- Start now. Open with one or two warm sentences: this is a conversation, not a quiz; my own examples matter more than being right; I can ask you to clarify anything. Then ask me to name one place where a flat map or floor plan works perfectly well for me, and one situation where I suspect a flat map might mislead.

TOPICS, IN ORDER
1. Flat where it works, curved where it doesn't. Explore my two examples. Ask why a flat plan works for a room or a town (the Earth's curve is tiny over short distances, so flat geometry is an excellent approximation). Then ask what changes for a long flight or a whole globe. Follow-up: on a ball, can two "straight" paths stay parallel forever?
2. The curved flight path. Ask why a flight between continents looks curved on a flat map. Draw out that the shortest route on a sphere follows a great circle, and that a flat map stretches the globe, so the true shortest path looks bent. If useful, mention that a triangle on a sphere, such as one with corners at the North Pole and two points on the equator a quarter of the way around from each other, can have three right angles, summing to 270 degrees.
3. Geometry you use without studying it. Ask what it means to me that my phone's GPS depends on a geometry most people never study. Explain only if needed: satellite clocks run fast by about 38 microseconds a day because of relativity, which describes gravity with a curved geometry of space and time, and this has to be corrected. Ask what other everyday tools might rely on mathematics I never see.
4. Closing. Ask me to say, in two sentences, which geometry I would use for my own town and which for the whole Earth, and why. Help me sharpen the reason, so I leave with a claim I could test by hand with a ball and a marker.

KEY POINTS TO KEEP ACCURATE
- On a sphere, a triangle's angles add up to more than 180 degrees; the larger the triangle, the larger the sum. Very small triangles come close to 180.
- Great circles are the shortest routes between two points on a sphere. On a sphere, any two great circles meet, so there are no parallel lines.
- In hyperbolic geometry, many lines through a point never meet a given line, and triangles' angles add up to less than 180 degrees.
- GPS satellite clocks run fast relative to ground clocks by about 38 microseconds per day; uncorrected, this would cause errors of more than 11 km in a day.
- Kissing numbers (how many equal spheres can touch one central sphere) are known exactly in only a few dimensions and are open in most.
- In 2025, an AI system, Google DeepMind's AlphaEvolve, set the record lower bound of 593 for the kissing number in dimension 11.

MISCONCEPTIONS TO CORRECT GENTLY
When one appears, name the accurate version briefly, then return to my examples.
- "Non-Euclidean geometry is just theoretical": GPS depends on relativity, whose mathematics is a curved, non-Euclidean geometry.
- "Flat maps show the true shortest routes": on a globe, great circles are shortest, and flat maps distort them into curves.
- "Euclid was wrong": Euclid's geometry is consistent and works very well on small, flat regions; it's one of several consistent geometries.

LIMITS
- No equations of relativity.
- Don't explain GPS beyond the points listed here.
- Don't settle which geometry describes the universe at the largest scales; if I ask, say it's a question for measurement.
- Don't introduce the history of geometry beyond a sentence.

TO FINISH
After my two-sentence answer, close in one short turn:
- Affirm one specific thing I worked out, in my own words where possible.
- Suggest one or two next steps that fit how the conversation went. Possible steps: trace a long flight route on a globe with a string and compare it with a flat map; look at how a triangle's angle sum changes with its size on a sphere; read about how GPS corrects for relativity; review what a great circle is.
- Restate my two-sentence answer on its own line, labeled "My geometry claim", so I can copy it.

Hands-on Activity 15 minOptional Done

Draw a Triangle with Three Right Angles

Overview

This optional activity lets you see, with your own hands, that a triangle's angles don't always add up to 180 degrees. On a flat sheet, they do. On the surface of a ball, the "straight lines" are great circles (circles whose center is the ball's center, like the equator), and triangles drawn with them have angle sums greater than 180 degrees. You'll draw triangles on paper and on a ball and compare.

What You'll Need

A ball or an orange
A marker, or a few rubber bands that fit snugly around the ball
A protractor, or a card with square corners (an index card or a business card works)
Paper and a ruler

Your Task

Draw one triangle on paper and two triangles on the ball, one large and one small. Check each corner's angle and add up the angles for each triangle. Then compare the three sums.

Steps

Draw a flat triangle and add its angles. On paper, draw any triangle with a ruler. Measure its three angles with a protractor and add them. If you only have a square-cornered card, draw a triangle with one square corner and estimate the other two.
Mark a pole and an equator on the ball. Pick a point on top and call it the North Pole. Draw, or stretch a rubber band around, the ball's widest circle halfway between the North Pole and the opposite point; that's the equator. Mark two points on the equator a quarter of the way around the ball from each other.
Join the three points along the surface. Connect the North Pole to each equator point with a line that runs straight down the ball, the way lines of longitude run on a globe. A rubber band stretched around the ball through the North Pole and the equator point will lie along the right path. The stretch of equator between the two points is the third side.
Check each corner and add the angles. Hold the square-cornered card against each corner of the ball triangle, keeping it as flat against the surface as you can. Note whether each corner matches the card's right angle, then add up the three angles.
Draw a much smaller triangle on the ball and compare. Draw a small triangle, a few centimeters across, anywhere on the ball, using short straight-looking strokes. Check its corners with the protractor or card, and compare its angle sum with the large one.

What to Expect

Your flat triangle's angles should add up to close to 180 degrees; a few degrees off is normal measurement error. On the ball, each corner of the large triangle should look like a right angle, so its angles add up to about 270 degrees. The small triangle on the ball should add up to only a little over 180 degrees, close enough that you may not be able to tell the difference. On a sphere, the larger the triangle, the further its angle sum rises above 180. Over a small patch, the surface is nearly flat.

Self-Check

When you're done, check that:

Your flat triangle's angles add up to close to 180 degrees.
Each of the three corners of your large ball triangle is close to 90 degrees.
Your small ball triangle's angle sum is close to 180 degrees.
You can say in a sentence why the size of a triangle matters on a sphere.

Nothing is uploaded. Write in your own notebook or document and keep it.

Knowledge Check 10 min Done

Curved space, GPS, and the packing of spheres

This ungraded knowledge check assesses your understanding of geometry beyond Euclid's flat plane and its uses today. You'll be asked about spherical and hyperbolic geometry, how GPS uses distances and relativity, the sphere-packing results, and kissing numbers as an open problem.

Note: Use this to test yourself, review the feedback on any questions you miss, and retry until you feel confident before moving forward.

5 questions · ungraded · retry as often as you like

Graded Quiz 30 min Done

Geometry

This graded quiz assesses your understanding of how geometry proves results from axioms, how non-Euclidean geometries differ from Euclid's, and where geometry is used and still open today. You'll be asked about geometry before and in Greece, axioms and proof in Euclid's Elements, similar triangles and trigonometry, the parallel postulate, spherical and hyperbolic geometry, GPS, sphere packing, and kissing numbers.

Note: Aim for a score of 80 percent or higher. If you score lower, use the feedback to review the topics you missed, then retake the quiz.

10 questions · target score 80% · 3 forms, rotated on each attempt