Arvind Borde

MATH 1: Week 8, Higher Dimensions and Curved Geometry

Dimension

(1) What are the coordinates of

1

NOTES:

(2) What are the coordinates of

2

NOTES:

(3) Plot the points (−1, 3) and (2,−3).

3

NOTES:

Here is a 3-d graph:

The coordinates of the point are (1, 1, 2).

4

NOTES:

In a 3-d graph, as well, the order in which

you move along the axes is clearly important.

Usually, axes are labeled x, y, z, and you

follow the convention that you proceed in

alphabetical order. We will, instead, label

the axes numerically (axis 1, axis 2, etc.) and

proceed in numerical order.

We do this in order to be able to discuss

higher dimensional graphs, where we would

otherwise run out of letters.

5

NOTES:

Higher Dimensional???????????????

Calm down, calm down.

The dimension of a problem in mathematics is

the number of variables needed to describe

it. There are many situations that need

more than 3 variables. These are “higher

dimensional” problems.

We ignore, for now, how we may visualize

higher dimensions.

6

NOTES:

We can write coordinates in any number of

dimensions. Here are examples:

2-d: (2,−3)

3-d: (−1, 2, 4.5)

4-d: (2, 3,−1, 2)

5-d: (3,−2, 1, 2,−4)

In general, a point P in n-dimensions will

have coordinates

(P1, P2, P3, . . . , Pn)

7

NOTES:

Distances between points

Slap on coordinates, and you get . . .

8

NOTES:

Coordinates:

P : (P1, P2) and Q : (Q1, Q2).

9

NOTES:

In the previous diagram

(4) How are d, a, and b related?

In terms of (P1, P2) and (Q1, Q2)

(5) What is a?

(6) What is b?

10

NOTES:

Putting it all together, we get

d2 = a2 + b2

= (Q1 − P1)2 + (Q2 − P2)

2

To get the distance d we

1) First get d2, then

2) Take its square root.

11

NOTES:

What are the distances between

(7) (1, 4) and (5, 1)?

(8) (−1, 3) and (1, 1)?

12

NOTES:

You can extend this to any number of

dimensions. Let P and Q be two points with

coordinates

P : (P1, P2, P3, . . . , Pn)

Q : (Q1, Q2, Q3, . . . , Qn)

The distance between them is obtained from

d2 =

(Q1 − P1)2 + (Q2 − P2)

2 + . . .+ (Qn − Pn)2then taking the square root.

13

NOTES:

(9) What is the distance between

(1, 2, 3) and (4, 1, 1)?

14

NOTES:

(10) What is the distance between

(−1, 3, 1, 1, 2) and (1, 3, 2, 1/2, 0)?

15

NOTES:

Visualization

Although visualizing higher dimensions is

not easy, it is possible to glimpse what some

aspects of higher dimensional objects might

be like.

We’ll look at cubes and ask ourselves how

they change as you go from one dimension to

the next.

16

NOTES:

2-d cube:

2-d → 3-d: Place two copies of a 2-d cube,

then connect matching vertices:

−→

17

NOTES:

How does the number of vertices and edges

change in this construction?

In 2-d, we have 4 vertices and 4 edges:

V2 = 4, E2 = 4

18

NOTES:

In going from 2-d to 3-d, we double the

number of vertices (we use two copies of

squares), and both double the number of

edges and add new edges. The new edges go

from old vertex to old vertex, so their number

is the number of old vertices:

V3 = 2V2

= 2(4) = 8

E3 = 2E2 + V2

= 2(4) + 4 = 12

19

NOTES:

3-d → 4-d: Place two copies of a 3-d cube,

then connect matching vertices:

Again, we double the number of vertices, and

both double the number of edges and add new

edges that go from old vertex to old vertex.

20

NOTES:

We getV4 = 2V3

= 2(8) = 16

E4 = 2E3 + V3

= 2(12) + 8 = 32

In general, going from n to n + 1 dimensions,

we have:Vn+1 = 2Vn

En+1 = 2En + Vn

21

NOTES:

(11) How many vertices and edges does a 5-d

cube have?

22

NOTES:

Curved Geometry

So far, we have looked at flat geometry. A

key feature of flat geometry is that the angles

of a triangle add to 180◦:

6 A+ 6 B + 6 C = 180◦

23

NOTES:

Now look at a triangle drawn on the surface

of a sphere:

(12) What does 6 A seem to be?

(13) What does 6 C seem to be?

24

NOTES:

Both angles seem to be 90◦. Therefore, on a

sphere6 A+ 6 B + 6 C > 180◦

This is true of any triangle that you draw on a

sphere with “straight lines” (lines of shortest

distance).

25

NOTES:

Surfaces where the angles of a triangle add

to greater than 180◦ are said to have positive

curvature.

Surfaces where the angles of a triangle add to

exactly 180◦ are said to have zero curvature,

or are called flat.

Surfaces where the angles of a triangle add

to less than 180◦ are said to have negative

curvature.

26

NOTES:

A saddle is an example of a space with

negative curvature:

27

NOTES:

Another approach to curved geometry is

through the distance formula. We saw that

the distance between a point with coordinates

(P1, P2) and one with coordinates (Q1, Q2) is√(Q1 − P1)2 + (Q2 − P2)2

This is the 2-d flat space distance formula.

When you use this formula it means that you

are working with flat space.

28

NOTES:

We’ll need the notation that “dx” means a

(very small) difference in the variable x. The

distance formula for flat space is then just the

sum of squares of coordinate differences:√d(first coordinate) 2 + d(second coordinate) 2

29

NOTES:

If you’re discussing the geometry of a sphere

you use a distance formula that defines that

geometry. Good coordinates for a sphere are

the latitude (represented by the Greek letter θ,

called “theta”) and the longitude (Greek letter

φ, called “phi”). If the radius of the sphere is

r, the distance formula is√r2dθ2 + r2 sin2 θdφ2

The curvature can be calculated from the

distance formula.

30

NOTES:

Curved geometry

is important for

many things – from

understanding the

structure of the Universe

to the new technology of

printing food.

31

NOTES:

Pasta 1: Spaghetti

x = 0.2 cos[u20π]

y = 0.2 sin[u20π]

z = v/10

u : 0 . . . 100, v : 0 . . . 100

32

NOTES:

Pasta 2: Linguini

x = 0.1 cos[u20π]

y = 0.2 sin[u20π]

z = v/10

u : 0 . . . 100, v : 0 . . . 100

33

NOTES:

Pasta 3: Orecchiette

x = 2v3 cos

[u75π]+ 0.3 cos

[2u15π]

y = 10 sin[u75π]

z = 0.1 cos[u3π]

+5(0.5 + 0.5 cos

[2u75π])4

cos[

v30π]2

×1.5(0.5 + 0.5 cos

[2u75π])5

sin[

v30π]10

u : 0 . . . 150, v : 0 . . . 15

34

NOTES:

Pasta 4: Fusilli

x = 6 cos[3u+10100 π

]cos[ v

25π]

y = 6 sin[3u+10100 π

]cos[ v

25π]

z = 3u/20 + 2.5 cos[v+12.5

25 π]

u : 0 . . . 200, v : 0 . . . 25

35

NOTES:

Pasta 5: Riso

x = 1.4 sin[

v80π]1.2

cos[u40π]

y = 2.5 sin[

v80π]1.2

sin[u40π]

z = 6 cos[

v80π]

u : 0 . . . 80, v : 0 . . . 80

36

NOTES:

Pasta 6: “Strano Riso”

x = 1.4 sin[

v80π]1.2

cos[u40π]

y = 2.5 sin[

v80π]1.2

sin[u40π]

z = 6 sin[

v80π]

u : 0 . . . 80, v : 0 . . . 80

37

NOTES: