Constructing Space A Do-It-Yourself Guide Rachel Maitra.

Constructing SpaceA Do-It-Yourself Guide

Rachel Maitra

Flat earth• On a map, the earth looks flat.• In real life, we see hills and valleys, but how do we

determine the overall shape of the earth?

Arguments for a spherical Earth

• Pythagoras: “Spheres rule!”• Aristotle:

– “There are stars seen in Egypt and Cyprus which are not seen in the northerly regions.”

– Self-gratitation should compact the Earth into a sphere– The Earth casts a round shadow on the Moon during a lunar

eclipsePythagoras

Aristotle

Eratosthenes: I can measure it

• In Cyrene, Egypt, there was a deep well. At noon on summer solstice, the sun shone directly into the bottom of the well.

Cyrene

Eart

hsunlig

ht

well

Eratosthenes: I can measure it

• In Cyrene, Egypt, there was a deep well. At noon on summer solstice, the sun shone directly into the bottom of the well.

• 800 km away in Alexandria, there was an oblisk which cast a shadow on that same day.

Eart

hsunlig

ht

well

Alexandria

obelisk

Eratosthenes: I can measure it

• In Cyrene, Egypt, there was a deep well. At noon on summer solstice, the sun shone directly into the bottom of the well.

• 800 km away in Alexandria, there was an oblisk which cast a shadow on that same day.

Eart

hsunlig

ht

well

Alexandria

obelisk

Eratosthenes: I can measure it

Eart

hsunlig

ht

well

Alexandria

obelisk

17.2 360

50Earth's circumference 50 800 km 40,000 km

(Today we know that it is ~40,075 km!)

Now the Earth’s shape is familiar…

…but space is mysterious.

Space in general relativity

• The universe can be described as a time part paired with a space part

( time)t

(space)

• Every “event” has a time coordinate and three space coordinates: (t, x, y, z) (time, longitude, latitude, altitude)

• Time is like the advancing filmreel. Space is whatever shape the stills show.

3-d slices

What shape can the stills have?

• Locally, space looks flat. We just have height, width, and depth.

• But just as Indiana looks flat and the Earth isn’t, the overall shape of space may be more complicated.

• But just as Indiana looks flat and the Earth isn’t, the overall shape of space may be more complicated.

• The surface of the Earth is a 2-dimensional sphere:

Cut out two disks, round them into hemispheres, and glue them along their boundary circles.

Definition of dimension

• Anything which can be made by cutting out pieces of a 2-d plane and glueing them together is considered a 2-dimensional surface.

• Space is 3-dimensional, which means we can imagine it by seeing what we can construct out of chunks of flat 3-dimensional space.

(note: these are hollow, not solid!)

Our building materials

2

3

n

←

Term

inolo

gy

→

Our building materials

2

3

n

←

Term

inolo

gy

→

How to construct 3-d spaces

• Example: a 3-d sphere → Space balls! Start with two

solid spheres:

How to construct 3-d spaces

• Example: a 3-d sphere → Space balls! Start with two

solid spheres:

Stick these two points together

How to construct 3-d spaces

• Example: a 3-d sphere → Space balls! Start with two

solid spheres:

• The outer surface of each ball is a 2-d sphere. Stick each point on the surface of the left-hand ball to its mirror image on the surface of the right-hand ball.

Stick these two points together

How to construct 3-d spaces

• Example: a 3-d sphere → Space balls! Start with two

solid spheres:

• The outer surface of each ball is a 2-d sphere. Stick each point on the surface of the left-hand ball to its mirror image on the surface of the right-hand ball.

Stick these two points together

N.B. We actually need 4 ambient dimensions to work in!

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

?

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

?

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

?

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

?

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

!

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

!

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

!

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

!

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

!

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

!

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

!

Difficult to visualize!• Since we only live in 3 dimensions, we need some tricks to

visualize the 3-d sphere.• Imagine you were stuck in a plane. How would you visualize a

2-d sphere?

Trick #1: Fly-by

!

What the planar being sees:

What the planar being sees:

What the planar being sees:

What the planar being sees:

What the planar being sees:

What the planar being sees:

What the planar being sees:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

If a 3-d sphere passed through our 3-d world:

…to a 2-d sphere of the same radius as the 3-d sphere (its “Equator”)…

We would see a succession of (hollow) 2-d spheres growing from a point (the “South Pole” of the 3-d sphere)…

…and then shrinking back to a point again (the “North Pole”).

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w A journey from pole to pole:

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w A journey from pole to pole:

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w A journey from pole to pole:

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w A journey from pole to pole:

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w A journey from pole to pole:

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w A journey from pole to pole:

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w A journey from pole to pole:

Trick #2: Jawbreaker

Imagine the two solid balls we use to make a 3-sphere are composed of thin multicolored layers.

We can use a gradual gradation of color to represent the fourth dimension needed to embed the 3-sphere.

0w 0.5w

0.8w 1w

0.3w 0.6w

1w

These two points are in fact the same.

A journey from pole to pole:

Need better tools for characterizing spaces!

• A sphere is a very simple shape, but already we have had to invent fancy tricks to visualize a 3-sphere. What about more complicated spaces? What about higher dimensions?

• We need topology.

• Topology is logically prior to geometry. It is the study of space in which we ignore measurements of distance, only noticing overall shape.

“A topologist can’t tell a coffee cup from a doughnut.”

Greek topos = space, logos = knowledge

Need better tools for characterizing spaces!

• A sphere is a very simple shape, but already we have had to invent fancy tricks to visualize a 3-sphere. What about more complicated spaces? What about higher dimensions?

• We need topology.

Greek topos = space, logos = knowledge

Can’t tell a coffee cup from a doughnut…• You might think this is a disadvantage…

…but in fact it allows us to focus on information about overall shape, ignoring irrelevant local bumps and dents.

• Similar to saying the Earth is a sphere, ignoring the fact that technically it has mountains and fault-lines, and is slightly oblate.

CONCEPT: HOMOTOPY EQUIVALENCE(Greek homo = same, topos = space)

Two spaces are homotopy equivalent if one can be smoothly deformed into the other (no tearing or glueing).

A doughnut is homotopy equivalent to a coffee cup.

Mathematics is about structure

…and about which structures of a given object to focus on. Different fields of mathematics study different types of structure.

Consider the real number line:

There are three main subfields of mathematics:

What does each of these see in the real line?

0 1 2 3 4 5 6123456

•Algebra•Analysis•Topology

In the eyes of the beholder…•Algebra

…sees the operations of addition and multiplication. It views the real numbers as entities which can be combined using

these operations to yield new entities which still belong to the set of real

numbers.

•Topology…sees the shape of the real line. It sees

that the real line is 1-dimensional, and that no looping paths are possible. It views the

real numbers as points in a space.

• Analysis…sees the microstructure of the real line. It views the real

numbers as entities which are ordered in a continuum.

The evolving universe

• One possible model (de Sitter space)

TIM

E

BIG BANG

BIG CRUNCH

Constant-time slice = 3-sphere

Shape of whole universe?

The evolving universe

• One possible model (de Sitter space)

TIM

E

BIG BANG

BIG CRUNCH

Constant-time slice = 3-sphere

Shape of whole universe?

→ 4-sphere!