THE�GEOMETRY�OF �EMPIRE
Constructing a model to help us understand
what happens as an empire expands
Imagine that a country
started in the corner
of a large continent.
(The continent is�much bigger than this screen,
but we are looking only
at the northwest part.)
Imagine that a country
started in the corner
of a large continent.
(The continent keeps going
farther south and east,
but we are looking only
at the northwest part.)
Michigan GLCEs
6-G4.4.1 Identify factors that contribute to conflict and cooperation
between and among cultural groups.
7-W312 Propose a generalization about the relationship between
geographic characteristics and development of early empires.
Next Generation Science Standards: Developing and Using Models
Modeling in 6–8 builds on K–5 and progresses to developing,
using and revising models to describe, test, and predict
more abstract phenomena.
MS‑ETS1‑4 Develop a model to generate data to test ideas � about systems, including inputs and outputs.
Question: What is a predictive model?
Answer: It is a set of assumptions
that can be “translated” into mathematical statements.
Then, you can “run” them for several time periods
in order to see what happens in a system
that is too complex to describe all at once.
Question: What does that mean?
Answer: Let’s build a model, to find out!
Question: What kind of model?
Answer: How about a model of empires,
to see how they grew (and fell) ?
(This model might help us understand
what’s happening in the world today!)
Assumptions
1. We can describe an area as a bunch of identical squares.
(This makes the math a lot easier.)
That’s it – it’s a simple model. Let’s see how it works,
and then fix it if we need to make it better.)
2. The empire gains 5 units of “loot”
(gold, silver, weapons, tools, food, slaves, etc.)
every time it captures a square of land.
3. It costs 2 units of “loot” to capture each “new” square
(to pay for troops, weapons, supplies, etc.)
4. It costs 1 unit of “loot” to rule each “old” square
that the empire already has
(to pay for government, police, roads, schools, etc.)
Remember – the empire started
in the northwest corner
of a big continent.
Here is our data table.
It will have a lot of numbers when we’re done,
but each one of them is really easy.
That’s the whole purpose
of a predictive model –
to do some simple tasks
in order to answer
a key question about
a complicated system!
For example . . .
In time period 1, the empire sends armies
to capture all 3 of the nearby squares.
For example . . .
In time period 1, the empire sends armies
to capture all 3 of the nearby squares.
How much loot
will the empire gain?
The empire gets 5 units of loot
from each captured square.
5 x 3 = 15 units total.
For example . . .
In time period 1, the empire sends armies
to capture all 3 of the nearby squares.
How much loot
will it cost to invade
the 3 new squares?
It costs 2 units of loot
to invade each square.
2 x 3 is 6 units total.
For example . . .
In time period 1, the empire sends armies
to capture all 3 of the nearby squares.
It costs only 1 unit
to rule over the
existing country,
which had only
one square of land.
What is the “profit”
for this expansion
of the empire?
L - cI - cR = P
15 - 6 - 1 = 8
In time period 2, . . .
the empire sends armies to capture
another “ring” of squares – 5 of them.
How much loot
will the empire gain?
What are the costs
to invade “new” squares?
What are the costs
to rule “old” squares?
What is the profit
in this time period?
In time period 3, . . .
the empire sends armies to capture
another “ring” of squares – 7 this time.
Here are the numbers
for this time period.
Check to make sure
they make sense.
Remember the assumptions:
Loot = 5 x number of “new” squares.
Costs to invade = 2 x # of “new” squares.
Costs to rule = 1 x # of “old” squares.
Profit = loot minus both costs.
In time period 4, . . .
the empire sends armies to capture
another “ring” of squares –
How many
this time?
How much loot
will the empire gain
and
what are the costs?
In time period 5, . . .
the empire sends armies to capture
another “ring” of squares –
How many
this time?
How much loot
will the empire gain
and
what are the costs?
In time period 6, . . .
the empire sends armies to capture
another “ring” of squares –
How many
this time?
How much loot
will the empire gain
and
what are the costs?
Here is the big question:
What will happen
in the next time period
if the empire stops invading new land?
What if it tries to capture more land?
What do you think
they could do about this?
Here are our conclusions:
If an empire relies on captured loot
to support its armies and its economy,
eventually, costs can grow
to become more than gains,
and the empire is in trouble.
To survive,
they have to figure out
how to increase their income
or reduce their costs.
Do you see how
making a simple model
can help us understand
the problems caused by
the “geometry of empire”?
Do you have any ideas
about how we could
change the assumptions
to make the model better?
Thought Question
Is the United States
an empire?
Consider these facts.
For more than a hundred years,
the United States government captured land
and used it to pay for things it wanted.
It gave grants of land near the frontier�to pay the soldiers in the Revolutionary War.
It gave land grants to the states�to pay for building schools and universities.
It gave land grants near the frontier�to pay Civil War soldiers (and their families).
It gave large grants of land to companies�who agreed to build railroads across the country.
And now, some politicians want to sell
the national parks and wilderness areas
in order to pay for the Federal debt.
The green areas on this old map
show where the U.S. government
gave millions of square miles
to the railroad companies.
This simple version of the land grant map
is in the clickable miniAtlas of the U.S.
You can use this electronic atlas
to investigate other conditions
in the areas of the railroad grants.
P.S. The railroads didn’t get
all the land in the red areas.
Often, they got every other section of land,
like the dark squares on a checkerboard.
Think about problems that could cause.
Thought Question
What happens to an empire
when there is no “new” land
to capture and use?