Batch 3 - Class 204 - Planiverse - Flatland

• (Moscow - 58) Stopped Clock - My clock stopped. I went to my friend's place, saw the time and waited for some time, and came back and set my clock right. How did I do it without knowing the time it took me to travel from my friends house back to mine
• Answer: I wound the clock before leaving home, so when I came back I knew the total time I was away. I watched my friend's clock when I reached and left my friend's house, so I know how long I was here. Hence I know my one way travel time. Adding that to time when I left my friend's house gives me the current time for my clock.

• Is the surface of earth 2-dimensional or 3-dimensional?
• What is dimension?
• It is 2-dimensional, since all points on the surface can be expressed using two independent values (eg: latitude and longitude). It is, however, embedded in a 3-D space

• How many dimensions does the world around us have?
• How do we know?
• Is it really three dimensional? Can we describe all coordinates with three dimensions? What are those?
• Are these only the spatial dimensions? Can you think about other kinds of dimensions?
• Color as an alternate spatial dimension? How would you use it?
• Time?
• When we see something with our eyes, how many dimensions do we see? By how many independent values can we describe anything that we see?
• Wait, but we do see a sense of depth. How?
• Stereo vision - but close one eye, and then see if you see sense of depth
• Hold a Rubik cube and you are able to see a cube and not just a hexagon. Why?
• Color (and shading) provides the perspective - indeed the "third" dimension
• How about a point on a given straight line - how many variables do we need to exactly describe the point?
• How about a point on a plane? On a sphere (back to earth?) In space?

• Explain the concept of Planiverse
• Flatland style (Edwin Abbott) - on floor
• Women are a straight line. Men can be polygons of different dimensions - workers are narrow triangles, middle class is equilateral triangles, Nobility ranges from hexagons onwards, and Priests are almost circular. Kids have one more dimensions than parents, except for workers who always remain triangles and over several generations of accomplishments, may approximate an equilateral. (Note the satire on caste and class structure here.)
• How do people recognize each other?
• By voice
• Feeling the corners 
• By sight - There is fog so that distance can be inferred. 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?)
• If color could help recognize, the sense of sight would remain under-developed, and hence the skills of upper classes were no longer valuable. But the more severe impact is that women could impersonate as priests by applying shading!
• How will people walk? How will they cross each other? Transportation?
• Lets build a house
• "Flatland" style pentagonal house

• How would a Flatlander encounter a Lineland?

• The king of course believes that his line is the entire universe
• What will happen as the Flatlander crosses the line? How will it appear/disappear?
• How would the Flatlander describe the notion of second dimension to the king?
• How do you convince a Planiverse resident that a third dimension exists?
• A "sphere" entering Planiverse would look like a circle (Priest) changing in size...
• What if the Flatlander was "lifted" into the third dimension. The Flatlander would be able to see the "inside" of all other Flatlanders (still on the Flatland) for example.
• How do we know that a fourth spatial dimension doesn't exist?
• How would you go from zero dimension (point) to one dimension (line) to two (square) to three (cube) to four and beyond?

• (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