Same Shape or Not?

Put a city's transit map next to a street map of the same city and they disagree. On the transit map, stations sit at even intervals along straight colored lines that bend only at neat angles. On the street map, the same stations bunch together downtown and spread far apart in the suburbs. Two stations that look a finger's width apart on one map can be a long walk apart on the other.

Yet the two maps agree on a great deal. They agree on which stations each line serves, in what order, and where two lines meet so you can change trains. Ask whether the two maps show "the same shape" and you get two honest answers: no and yes. Which answer is right depends on what you mean by "same."

Geometry gives the first answer. Geometry is the branch of mathematics that studies shape, size, and position, so it cares about distances and angles. A street map drawn to scale keeps the ratios of distances, and that lets you compute real lengths from it. One of the oldest tools for that job is the Pythagorean theorem. In a right triangle, the square on the side opposite the right angle equals the sum of the squares on the other two sides (Euclid, Elements I.47). Measure two legs of a right triangle on a scaled map and you can work out the straight-line distance between two points.

Topology gives the second answer. Topology is the branch of mathematics that studies the properties of a shape that survive continuous stretching and bending, as long as nothing is torn apart or glued together. Distance and angle don't survive that treatment. Connections, order, and holes do. The word itself is a nineteenth-century coinage: Johann Benedict Listing used "topology" in print in 1847 (O'Connor and Robertson 1996c). A transit map is close to a topological drawing. It throws away distance on purpose and keeps the connections exact.

That leaves a practical problem. In the project, a city is redesigning its transit map, and a resident group has asked you why the new map "isn't to scale" and whether that matters. Your report is your answer, and it has to settle three doubts:

• What does each map get right?
• What does each map distort?
• Which questions does each map serve?

If your answer dismisses distance, riders will plan walks they can't make. If it insists on scale, the map gets crowded and hard to read.

You'll write a report of 600 to 900 words that compares two maps of one place, or two objects. You'll make two geometric measurements, state two topological properties, pose one question only geometry can answer and one that topology answers as well, and end with your judgment of when each view is the right one.

This content reflects the field as of October 2026.

References

• Euclid. Elements. Edited and translated by David E. Joyce, after T. L. Heath's edition. Worcester, MA: Clark University, 1996–2013.
  • Free: Euclid's Elements, Book I (Joyce)
• O'Connor, J. J., and E. F. Robertson. 1996c. "A History of Topology." MacTutor History of Mathematics Archive, University of St Andrews. Last updated May 1996.
  • Free: MacTutor: A history of Topology

Deliverable Overview

You'll write one report, 600 to 900 words long, for readers who aren't mathematicians. It compares what geometry measures and what topology keeps in two maps of the same place, or in two objects that are "the same" in one sense and not in another. A transit map and a street map of one city are the most direct choice. Two objects work as well: a photograph and an enlarged print of it, for example, or a rubber band lying slack and the same band stretched around a box.

The report has five required components:

1. The two maps or objects, described or sketched
2. At least two geometric measurements made on one of them
3. At least two topological properties stated for both
4. Two practical questions: one that only the geometric view can answer, and one that the topological view answers as well
5. A conclusion on when each view is the right one

Short headings for each component make the report easier to follow. Citations are welcome but not required. If you use them, give the author, year, title, and where to find the source, for example: O'Connor and Robertson, 1996, "A History of Topology," MacTutor History of Mathematics Archive (free online).

Detailed Requirements

The Two Maps or Objects

Describe each map or object well enough that a reader who can't see it could picture it. For a map, say what place it covers, who made it, what it's for, and whether it shows a scale. For an object, say what it's made of and how the two versions differ. A rough sketch with labels can replace some of the description. Name the specific features you'll use in the rest of the report, such as three stations, two street corners, or a hole in an object.

Geometric Measurements

Make at least two measurements on one of the maps or objects, and show your working. Geometry, the study of shape, size, and position, is the view that cares about lengths and angles, so these measurements should produce a length, a distance, an angle, or a ratio.

Two kinds of measurement fit well. The first uses the Pythagorean theorem: in a right triangle, the square of the longest side equals the sum of the squares of the two shorter sides (Euclid, Elements I.47). Suppose a street map has a scale of 1 to 10,000, so 1 cm on the map stands for 10,000 cm, or 100 m, on the ground. Two corners lie 6 cm apart east to west and 8 cm apart north to south. The straight-line distance on the map, d, satisfies

so d = 10 cm on the map, which is 1,000 m on the ground. Walking along the grid of streets takes 6 + 8 = 14 cm of map, or 1,400 m. The straight line is the shortest possible route, not the one you'll walk.

The second kind uses similar triangles. Two triangles are similar when they have the same angles; their matching sides then all share one ratio, called the scale factor. A scale bar on a map, or a known length on an object, gives you that ratio. If a scale bar 2 cm long is labeled 500 m, every map length converts to ground length at 500 m per 2 cm, or 250 m per cm. For two objects, measure the same feature on both, such as the width of a photograph and of its enlargement, and divide.

Topological Properties

State at least two topological properties, and say whether each holds for both maps or objects. Topology is the study of the properties of a shape that survive continuous deformation. A continuous deformation is any stretching or bending that doesn't tear the shape apart or glue separate parts together. A property that survives every such deformation is called a topological invariant.

For maps, good candidates include which stations or places are connected to which, the order of stops along a line, where two lines meet, and whether a route forms a closed loop. For objects, consider the number of holes, whether the object is in one piece, and whether a closed loop drawn on it can be shrunk to a point without leaving the surface. Check each property on both items. The interesting finding is often that the two maps disagree about distance and agree about every property on your list.

Two Practical Questions

Write two questions a real person might ask. The first must need geometric information, so neither map's connections alone can answer it: "How long is the walk from the station to the library?" The second must be one the topological view answers as well, so the connections alone are enough: "Can I get from the airport to the stadium without changing trains?" For each question, say which map or view answers it and why.

Conclusion

End with your judgment of when each view is the right one. Tie it back to the resident group's question: does the new map's departure from scale matter, and for which riders and which tasks? A strong conclusion names a situation in which the topological view is enough, one in which it isn't, and what a map maker could add to cover both.

Success Criteria

Your report meets the requirements when:

• Both maps or objects are described or sketched
• Two geometric measurements are shown with their working
• Two topological properties are stated for both
• One question needs geometry; another can be answered topologically
• Every technical term is defined on first use
• The conclusion says when each view is the right one

Step-by-Step Guidance

Step 1: Choose two maps or two objects. Pick two maps of one place, such as a transit map and a street map, or two objects that are "the same" in one sense and not another. Free online maps of your own town work well. Check that at least one of your two items has a scale or a known length you can measure against.

Step 2: Make two geometric measurements. Work on the item that has a scale. Measure with a ruler, write down each map length, convert it with the scale, and show every line of the calculation. Round sensibly and give units.

Step 3: List the topological properties. Write down at least two properties, and check each on both items. Keep a note of which question each property could help answer, because the report needs that link.

Step 4: Write the two practical questions. Draft one question that needs distance, angle, or area, and one that connections or order can settle. Test the second one: if you deformed the map without tearing it, would the answer stay the same? If it would, the question is topological.

Step 5: Draft the report for a non-specialist. Write for the resident group, not for a mathematician. Define each technical term the first time you use it, including geometry, topology, scale, and invariant. Keep the calculations short and readable.

Step 6: End with your judgment. Say when each view is the right one, and for whom. Give a reason, not a slogan.

Resources

• Euclid's Elements, Book I (Joyce): David E. Joyce's online translation, with diagrams. Proposition 47 is the Pythagorean theorem.
• MacTutor: A history of Topology: a short history of the field, from Euler's polyhedron formula to Poincaré.
• Your own notes. You may draw on notes or practice work of your own on the Pythagorean theorem, similar triangles, and properties that survive stretching.

Common Pitfalls

• Treating "not to scale" as an error. A transit map that ignores distance has made a choice. It trades accurate lengths for readable connections. Your report should say what that choice gains and what it costs, not treat it as a mistake to fix.
• Listing properties without saying which question each answers. "The lines meet at Central Station" is a property. It becomes useful when you add that it answers "Where can I change trains?"
• Measuring on a map without its scale. A length on paper means nothing until you know what it stands for. If a map has no scale, as many transit maps don't, measure on the other map.
• Confusing everyday "same shape" with either technical sense. In everyday speech, "the same shape" usually means similar: the same angles, possibly a different size. Topology's sense is far looser, and geometry's congruence, same shape and same size, is stricter. Say which sense you mean each time.

References

• Euclid. Elements. Edited and translated by David E. Joyce, after T. L. Heath's edition. Worcester, MA: Clark University, 1996–2013.
  • Free: Euclid's Elements, Book I (Joyce)