Algebra, Geometry, and Topology: What's The Difference?

Transcript

[Music] most people understand that science is a broad term describing many different fields of study like chemistry biology and physics [Music] what makes these scientific fields different is not necessarily that they study different objects but that they use different techniques to study the objects just like science mathematics is also a broad term that describes many different fields of study like algebra geometry and topology and many others however many people don't understand how these mathematical fields are different so let's take a closer look at how a geometry a topologist and an algebra study a circle to a geometry a circle is made of a rigid material and it has a perfectly round shape from this geometric perspective we can study the curvature of a circle as well as reflectional and rotational symmetries of a circle [Music] from a topologist point of view if you look at a circle from different angles it no longer looks perfectly round it might as well be made from a flexible material from this topological perspective a circle can take on many different shapes and forms so we can learn about knotting and braiding of circles from an algebraic point of view a circle is a collection of separate points along with a rule for connecting the points we can choose different connection rules to describe the same circle and from this algebraic perspective the circle can be used to visualize non-standard arithmetics although these fields of study are very different the greatest mathematical discoveries happened in the different fields of mathematics work together [Music] you