The Greeks were not interested in creating maths just for its own sake, but to give ordinary people confidence to participate in democracy. Greek philosophers and matematicians were interested in looking for things that were always true or sometines true and deciding the conditions which made a sometimes into an always or an always into a sometimes. Geometry is about working with pictures in your head, knowing what is fixed and what it is possible to move. A drawing made on papyrus is only one snapshot (a particular) representing an infinity of possible pictures (the general).
Much of Euclidean geometry can be approached as thought experiments. In the present day we have moving images on a computer but the Ancient Greeks had to create these moving images in their heads. What teachers need to convey to students is the power of working in this way.
One of the objectives of the "design a museum" summer school was to convey Euclid's actual methods of teaching, which included "thought experiments" - working with pictures in your head. For the demonstration, the teacher, in role as Museum Curator, further takes on the part of Euclid. In this double role he talks to the student Museum Designers as Euclid might have addressed his own students students:
"Good morning, everyone. Our Geometry class today will be another thought experiment. Please si comfortably. Place your hands on your knees. Close your eyes. Imagine a rectangle. Make sure you can fix it in space. Imagine a circle around the rectangle. Now fit the rectangle into the circle so that all four corners lie on the circle. Check round the four corners. Is each one touching the circle? Corner 1? Corner 2? Corner 3? Corner 4?
"Now keep your circle the same size but make the rectangle grow thinner and longer, but keep the four corners fixed on the circle. Now make the rectangle grow fatter and shorter, still keeping the corners on your circle. Make it grow and shrink inside your circle. Now stand back from your rectangle and circle picture and ask yourself if it is possible to draw a parallelogram in a circle with all its corners on the circle. Try really hard with the parallelogram. Try to make those corners reach the circle. Is it possible?
"Now open your eyes. Is it true that every rectangle will fit inside the circle? Remember that when we draw in the sand, we draw only one picture; when we draw in our imaginations we can make many pictures."
Students then discussed this as a possible way of demonstrating to museum visitors the importance to the Greeks of working with the imagination. Was it clear? Would people who didn't know anything about geometry understand it?
Adapted from 'No Royal Road to Geometry'