Some "Unit Circles" in different metrics... 

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can be equipped with many different metrics. Each of these yields the same topology, but their geometries are quite different from each other. 

Suppose that and belong to . Then for we have is the -metric for . If you take the limit as , you will get  the ∞-metric. 

The first two of these metrics have special names. is the taxi-cab metric (distances are sums of differences of coordinates - like having to travel down perpendicular streets). is the familiar Euclidean metric (our usual distance formula). It turns out that the topologies generated by all of these metrics yield the same topology.  

In the sheet below I generate plots of the set the unit circle/sphere centered at the origin. The first pair of plots deals with and . The second pair of plots deal with and . 

>	restart; with(plots):

 

Unit circles... 

>	circle := p -> (abs(x)^p+abs(y)^p)^(1/p)=1: circle(p);

 

(1)

 

>	for k from 1 to 10 do  cPlot[k] := implicitplot(circle(k),x=-1.1..1.1,y=-1.1..1.1,numpoints=5000,title=k): end do: cPlot[11] := implicitplot(max(abs(x),abs(y))=1,x=-1.1..1.1,y=-1.1..1.1,numpoints=5000,title=infinity):

 

>	display(seq(cPlot[k],k=1..11),insequence=true);

 

>	display(seq(cPlot[k],k=1..11),title="1,2,...,10,infinity");

 

Unit spheres... 

>	sphere := p -> (abs(x)^p+abs(y)^p+abs(z)^p)^(1/p)=1: sphere(p);

 

(2)

 

>	for k from 1 to 10 do  sPlot[k] := implicitplot3d(sphere(k),x=-1.1..1.1,y=-1.1..1.1,z=-1.1..1.1,numpoints=15000,title=k,style=surface): end do: sPlot[11] := implicitplot3d(sphere(1000),x=-1.1..1.1,y=-1.1..1.1,z=-1.1..1.1,numpoints=15000,title=infinity,style=surface):

 

>	display(seq(sPlot[k],k=1..11),insequence=true);

 

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