Batch 2 - Class 26 - Planar Geometry and Planiverse

• Let kids demonstrate their tilings

• MartinShCol 5.9 - Metal sheet has two semicircular ends and a square middle, square side and semicircle dia being 2 units. You cut a circle at centre of 2 unit dia. What is the area of remaining metal.
• Answer: 4
• MartinShCol 5.4- An equilateral triangle and regular hexagon have perimeters of same length. If triangle has area of two sq units, what is the area of hexagon?
• Answer: 3

• Explain the concept of Planiverse
• Flatland style (Edwin Abbott) - on floor
• Astria (Henton/ Arde Dewdney) Style - on wall, i.e. gravity
• How do people recognize each other?
• Arde - Full shapes and features?
• Flatland - Just polygons and sense of depth through lighting? Show a cube as example of how we use the same mechanism in 3D world
• They could still do stereo vision
• Color - banned in Flatland (why?), Ok in Arde
• How will people walk? How will they cross each other? Transportation?
• Pits with movable cover?
• Helium Balloons?
• Could one use wheels? Tanker style?
• People's bodies
• 

• Can't have tubes - Zipper organs

• Lets build a house
• "Flatland" style pentagonal house

• More sophisticated underground housing 

• Swing Stairs
• Water Tank
• Collapsible furniture
• Lights
• No nails and saws
• (Note that  the fundamental assumption in above two with respect to "ground" is different)
• Games people play
• One dimensional checkers
• Two dimensional chess as a more sophisticated design
• Orbit of the Planiearth?
• Gravity does not decay by square of distance but by distance => circular orbits
• No seasons
• Sophisticated Machines
• Example of Engine 

• How do you convince a Planiverse resident that a third dimension exists?
• How does a Planiverse resident convince a single dimension world resident that a second dimension exists?
• How would we convince a Planiverse resident of third dimension?
• How do we know that a fourth spatial dimension doesn't exist?

• Shakuntala - 81 - Divide a crescent into 6 parts by two straight lines
• MartinShCol 5.16 - form an enclosed space of 4 unit area using twelve matches - what can we do for all possible areas?
• Answer: Triangle 3x4x5 solution, Star solution

The Planiverse: Computer Contact with a Two-Dimensional World, by A. K. Dewdney