When we think about round objects, circles and spheres usually come to mind. But what exactly makes a sphere round? It’s all about how well a shape can enclose a large volume. This is known as “sphericity.” A sphere is the shape that holds the most volume with the least surface area, making it the most “spherical” shape.
However, roundness isn’t just about volume; it’s also about rolling. A round object can roll smoothly, like a wheel or a ball bearing. For an object to roll smoothly, it needs to have the same width from one side to the other. Surprisingly, there are many shapes that aren’t circular but still have this property. These are called “shapes of constant width.”
One interesting example is the Reuleaux rotor. This shape is made from segments of three circles that intersect at their centers. Because of this design, every point on its side is the same distance from the opposite corner, allowing it to roll smoothly. Amazingly, the Reuleaux rotor can even turn smoothly in a square hole!
But don’t get too excited about using a Reuleaux rotor as a car wheel. Since it’s not a constant distance from the axle, it would make for a bumpy ride. The points on a Reuleaux rotor are sharp, which seems to go against the idea of roundness. In geology, stones with sharp edges become “rounded” as they wear down. So, we can say a Reuleaux rotor is round but not rounded.
Some British coins, like the 20 and 50-pence pieces, are both round and rounded. They are shapes of constant width, which means they have a cool, unique look but don’t get stuck in coin machines. Isn’t it interesting that the word “rouleau” means “roll” in French, and Franz Reuleaux invented a rolling rotor?
In conclusion, roundness is a fascinating concept that goes beyond simple circles and spheres. Whether it’s about enclosing volume or rolling smoothly, there’s a lot more to explore in the world of shapes!
Use clay to mold different shapes, such as a sphere, cube, and cylinder. Measure and compare their surface areas and volumes to understand why a sphere is the most “spherical” shape. Discuss how these properties relate to the concept of sphericity.
Create cardboard cutouts of various shapes, including a circle and a Reuleaux triangle. Roll them along a flat surface and observe which shapes roll smoothly. Discuss why some non-circular shapes can still roll smoothly due to their constant width.
Draw and cut out a Reuleaux rotor using paper or cardboard. Test its ability to roll through a square hole. Reflect on how its unique properties allow it to roll smoothly despite not being a traditional circle.
Collect different stones and observe their shapes. Discuss how natural processes cause sharp-edged stones to become rounded over time. Compare these natural shapes to the Reuleaux rotor and discuss the differences between being “round” and “rounded.”
Examine various coins, focusing on those with constant width like the British 20 and 50-pence pieces. Discuss how their shape prevents them from getting stuck in machines and explore the practical applications of shapes of constant width in everyday objects.
Roundness – The quality of being circular or curved in shape. – The roundness of a circle can be measured by its circumference and diameter.
Sphericity – The degree to which an object approximates the shape of a sphere. – The sphericity of a basketball is important for its performance during a game.
Volume – The amount of space occupied by a three-dimensional object, measured in cubic units. – To find the volume of a cube, you multiply the length of one side by itself three times.
Rolling – The action of an object moving smoothly along a surface by turning over itself. – The rolling of a cylinder down a slope is an example of rotational motion in physics.
Width – The measurement or extent of something from side to side. – The width of a rectangle is one of the dimensions needed to calculate its area.
Shape – The external form or appearance of an object defined by its outline or surface. – The shape of a triangle can be classified based on its angles and sides.
Reuleaux – A curve of constant width, which is not a circle, often used in geometric constructions. – A Reuleaux triangle is a shape that can rotate smoothly inside a square while maintaining contact with all four sides.
Coins – Flat, typically round pieces of metal used as money, often used in geometry to illustrate circular shapes. – When studying geometry, coins can be used to demonstrate concepts like circumference and area of circles.
Geometry – The branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. – In geometry class, we learned how to calculate the angles of a polygon.
Smooth – Having an even and regular surface or consistency; free from perceptible projections, lumps, or indentations. – A smooth curve in geometry is one that does not have any sharp angles or discontinuities.
