Lineland Today we’ll be travelling to a new world: the one-dimensional land of Lineland. This is a part of Lineland, a world where everything is one dimensional. Here, they have no idea of a triangle, a square, a circle, a rectangle, or any other 2D shape you can name. Now imagine some inhabitants of a nearby world of Flatland come to Lineland. Linelanders have never had visitors from other dimensions and they see some strange things that day. As Sam the square and Cara the circle enter Lineland the Linelanders see only the part of them that intersect with their world. For example, with Sam they only see a line segment passing through at any time. The length of the segment is the same throughout the process. What would Linelanders see as Cara the circle passes through Lineland? (Hint: What they see changes as Cara passes through.)

As Sammy passes through: What Linelanders see:

As Cara passes through: What Linelanders see:

Fill in the last two lines on this table.

2.) What if Sam and Cara passed through Lineland at a different angle? Would Linelanders see something different? Draw what Linelanders would see as Sam passed through Lineland. Would Linelanders see anything different as Cara passed through Lineland at a different angle? Why?

As Cara passes through: What Linelanders see:

As Sammy passes through: What Linelanders see:

Do the same for the following shapes:

Look at the series of pictures below. What letter corresponds to this series? Do these pictures come from “A”, “C”, “E” or “W”? Do these pictures come from “Q”, “H”, “R” or “M”?

Do these pictures come from “Z”, “S”, “Y” or “L”? Do these pictures come from “P”, “O”, “C” or “B”?

Flatland

Now imagine some shapes passing through Flatland. Let’s take a cube for example:

What would Flatlanders see as a cube passes through their world? Draw what they would see. (Hint: it’s a

simple shape)

Now let’s explore what Flatlanders see as a ball passes through Flatland. Describe in words what the Flatlanders would see as the ball passes from one side of Flatland to the other.

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How about a square pyramid passing through the plane with its bottom face parallel to the plane? Draw what the Flatlanders would see at each time in the boxes below. Draw what they would see at the first instant, one quarter through, halfway through, three quarters through, and right before the tetrahedron exits.

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What would happen if the square pyramid approached Flatland with one of the other faces first? Would the cross-sections (pictures) change? Match each of the below vases with one of the series of pictures. 1

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Top Left: Vase 1 Top Right: Vase 2 Bottom Left: Vase 3