MARTIN GARDNER AND "FLATLAND"

Four times in major essays in the Scientific American, Martin Gardner specifically treated the dimensional analogy, as introduced by Edwin Abbott Abbott in the 1884 classic "Flatland".  In the first of those pieces, entitled "Flatlands", he presented a puzzle that has surfaced once again as
a central feature of the latest animation of that book, "Flatland the Movie", where the two-dimensional protagonists confront a three-dimensional
cube with its center fixed in their plane as it rotates continually in three-space.  Of course Flatlanders can only see the slices of this object
in their universe, sometimes a square, other times a rectangle or a rhombus or, most surprisingly, a hexagon.  Years ago in one of his Scientific
American columns, Martin Gardner challenged his readers to determine which of these "central sections of a cube" would have the greatest area.  The somewhat surprising answer was given in the next issue.

Now that the same problem reappears in animated form in this new version of the Flatland story, teachers have a new opportunity to explore questions about these central slices with their students.  As part of this gift for G4G8, we present selections from pedagogical materials being prepared to go along with the DVD of this animation, for use by teachers in middle school, secondary school, college and university, and for general audiences.

Special thanks go to Brown University undergraduate assistants Michael Schwarz and Tan Van Nguyen who created the Java animations using locally developed software created over the years by other undergraduates, supported by funds from Brown University and from the National Science Foundation.

HEX EXPLORES THE AREA 33H CUBE SLICES
  
    When Hex goes to Area 33H to see the Old Ruins for herself, she says, "I was right!" Visitors many centuries ago had left a sign so that people in Flatland might come to know the existence of the Third Dimension. Although Hex could not appreciate it when she first came to that monument, she was seeing a cube rotating in space about its center.  Of course since she was in the plane, she couldn't see the whole cube all at once, but she could see the slices of the cube sitting in Flatland in front of her.  By observing all of those slices, she could begin to understand quite a bit about this mysterious cube.

    Hex could keep track of different slices, and we can too, especially if we use color. In the movie, all of the square sides of the cube have the same color. It will help us to describe our observations if we color the six sides of the cube, with two opposite square colored red, two opposite sides colored yellow, and the remaining two opposite sides colored blue.  
colored cubecolored cube2

    As long as we are coloring the square faces of the cube, we might as well color the edges too.  An edge where a red square meets a blue square we can color purple, and similarly we can color orange each edge where a red square meets a yellow square.  What color should an edge where a blue square and a yellow square meet?  How many of the edges will be colored purple?  How many orange?  How many green?  How many segments are there all together?


    We are now ready to study the horizontal slices of the colored cube.  The easiest slice to understand is a square.  If we hold a cube so that one of the square faces is horizontal, then the slice of the cube that lies in the horizontal plane through the center will be a square.  If the horizontal squares are red, then the blue and yellow squares are all sliced the same way, producing a square with two blue edges and two yellow edges. What are the other possibilities?

horizontal slice1Horizontal slice2

    Now we want to see what happens if the cube starts to rotate.  If we start with a cube with red squares on the top and bottom and with yellow square in the front and back, then we can rotate it so that the yellow squares stay in the front and back.  Each of the yellow squares rotates about its center.  At the start of the rotation, the slices of these horizontal squares will be yellow segments with green endpoints.  The other two edges of the horizontal slice of the cube are blue with green endpoints.  As the cube rotates about the centers of the two vertical yellow squares, we obtain a quadrilateral that is no longer a square but a rectangle with two blue segments and two yellow ones.  The two blue segments stay the same length while the yellow segments grow larger until the square has rotated one-eighth of the way around and the horizontal slice segments are now the diagonals of the yellow squares.  At this point, the horizontal slice of the cube is a rectangle with two long yellow segments and two purple segments of the original cube, where a red square meets a blue square. 

Rotating cube1

Rotating cube2

      If we continue to rotate, then we will not slice the blue squares any more, and we now start to slice the red squares.  We get rectangles with two red segments that stay the same length while the yellow segments are getting smaller.  By the time that the two blue squares are horizontal, the central slice of the cube is a square, with two sides red and two sides yellow.  What will happen if we continue?  When do we get back to the starting point?



      If we stop the rotation when two blue squares are horizontal, the slice will be a square with two segments yellow and two red.  What will happen if we not rotate the cube so that the two red squares rotate about their centers?  

Rotating cube3



      We know that the perimeter of the rectangle is the sum of the lengths of its four sides, and the area of a rectangle is the length of one pair of sides multiplied by the length of the other.  Which of the horizontal rectangles has the smallest perimeter and which has the largest?  What about the areas of the rectangular horizontal slices as the cube rotates?

      Describe what happens if we start with red squares horizontal and rotate about the centers of the two blue vertical squares.  What can we say about the shapes and colors of the slices, and what about their perimeters and areas?  (It is possible to estimate the perimeter and the area by measuring two sides of each rectangle. Students who know about the Pythagorean theorem can find the perimeter and area of the largest of these slices in terms of square roots.  Students who know trigonometry can express the perimeter and area of a rectangular slice in terms of sines and cosines of angles of rotation.)

      We continue our investigation of the central slices of a cube by stopping when the horizontal slice is a rectangle with two purple edges and two longer yellow segments.  We now rotate the cube so that a diagonal of this cube stays fixed. Which way we rotate makes a difference.  One way gives quadrilaterals with two yellow segments and two blue ones and the other gives quadrilaterals with two yellow segments and two red ones.  Consider first the case where two segments are blue and two are yellow.  Since the two yellow squares are parallel in space, when they are cut by the horizontal plane, we have to get two parallel yellow segments.  Similarly the horizontal slices of the two parallel blue squares will be two parallel blue segments.  The quadrilateral slice therefore has to be a parallelogram.  At the beginning of the rotation, the yellow segments of the parallelogram are longer than the blue segments, and when the horizontal plane goes through the midpoints of a pair of green edges of the cube, the yellow segments and the blue segments have the same length.  The slice of the cube is then a rhombus.  Even though this non-square rhombus would be considered an irregular figure in Flatland, Hex can appreciate its symmetry. 




     What is larger, the perimeter of the rectangle or the perimeter of the rhombus?  What about the areas of these figures?
     What happens to the rhombus as we continue the rotation about the diagonal of the horizontal rectangle? 

Diagonal rotation1

Diagonal rotation2

    Naturally Hex is particularly interested in slices that are hexagons.  Since the cube only has six square sides, we cant get more than six edges in any slice.  Moreover, by the argument given above, the segments in the central slices come in pairs of parallel edges.  As it happens, we can get hexagonal slices by can get some of those by rotating a horizontal rhombus around its short diagonal.  If the rhombus has two blue segments and two yellow ones, then as the cube rotates, two small blue segments will appear.  The slice has six edges, with opposite segments parallel and colored the same.  At one position as the cube rotates, the horizontal slice has all six segments exactly of the same length, and with all angles equal.  We get this perfectly symmetrical hexagonal slice when the cube is positioned so that one of its long diagonals is vertical. For Hex, that is the most satisfying discovery--a slice of this mysterious cube from the third dimension can be a square, a rectangle, a rhombus, or, nicest of all, a hexagon like herself.



Hex4

    There is another way of getting to the regular hexagonal slice by starting with the square slice that has two yellow segments and two blue segments and four green vertices at the midpoints of the vertical green edges of the cube.  If we rotate about a diagonal of this square, then at the beginning, we have to get parallelograms, by the argument given above, and since the blue and yellow squares are treated equally in this rotation, we have to get rhombi, all the way up to the point where we have the largest rhombus, with two points at opposite corners of the cube.  If we continue the rotation, then small red segments appear, as in the previous exploration, and ultimately these have the same length as the blue and yellow segments.  At this point, all of the segments have the same length and all of the angles are equal!

Hex1

Hex2

Hex3

 

    For those of us who live in three-space, another way of studying the slices of cubes by planes is to keep the cube fixed and rotate the plane.  This change of viewpoints is relates to the idea of relativity since we are interested in the relative positions of the cube and the plane.  We can determine the position of a plane through the origin by seeing how it intersects a unit sphere---More later!



SLICING CUBES AND HYPERCUBES IN "FLATLAND: THE MOVIE"

     We have analyzed the slices that arise when the horizontal plane through the origin slices a cube rotating about its center that is fixed at the origin.  Our approach up to this point has been synthetic, without specific attention to the formulas that the computer uses to determine the slices and to display them.  We now start with some elementary analytic geometry of lines in the plane to show how to find slices of a square, and then how to extend this approach to find slices of a cube in three-dimensional space.  As we will see, this approach generalizes even further to allow us to investigate the slices by our three-dimensional universe of a hypercube rotating in four-dimensional space with its center fixed at a point in our world.

SLICING SQUARES BY LINES IN THE PLANE

     Slicing squares throught the center is an easy exercise. Whenever we slice a square with vertices (+-1,+-1), we get a segment meeting the boundary of the square at two opposite points.  If the horizontal edges, with y = +-1, are red and the vertical edges, with x = +-1 are blue, then the four vertices are colored purple.  Any non-vertical line ax + by = 0 will have slope m = -a/b.  The endpoints of the slice segment will be a pair of purple vertices if m = +-1, a pair of blue vertices if |m| < 1 and a pair of red vertices if |m| > 1.

     More generally a line ax + by = e will meet the four lines that determine the square in four points: (1,(e-a)/b). (-1,(e+a)/b),((e-b)/a,1) and ((e+b)/a,-1).  Exactly two of these four points will be on the boundary of the square.  The only way for one of the points to be purple is if either (e-a)/b or (e+a)/b is +-1.  In particular both endpoints will be purple if e = 0 and a = +-b.  If |(e-a)/b| and |(e+a)/b| are both less than 1, both endpoints will be blue, and if |(e-b)/a| and |(e+b)/a| are both less than 1, then both endpoints will be red.  We will have one endpoint blue and one red if exactly one of the values |(e-a)/b| or |(e+a)/b| is less than 1, in which case just one of the values |(e-b)/a| or |(e+b)/a| is less than or equal to 1.

SLICING CUBES BY PLANES IN THREE-DIMENSIONAL SPACE

     We can use this information to describe the slices of a cube by a plane through the origin.  If the cube in 3-space has vertices (+-1,+-1,+-1), then the slicing plane ax + by + cz = 0 meets the top square with z = 1, if at all, either in a vertex, in an edge, or in a segment.  If the top square is colored red, with edges colored purple if y = +-1 or orange if x = +-1.  The intersection of the plane with the top red square will be either the empty set or a vertex of the square or an edge of the square or a segment joining two vertices of the square face. Whatever the intersection is with the top square face, that same intersection occurs on the opposite bottom face, i.e. if (x,y,1) is in the intersection on the top, then (-x,-y,-1) is in the intersection on the bottom.  Moreover the segments on the top and bottom faces will have the same length and be parallel.

     A similar analysis shows that the central slices of the cube will meet the two vertical blue faces, if at all, each in a point, or a pair of opposite edges of the cube or in a pair of congruent parallel blue segments, and the same is true for the intersection with the pair of vertical yellow squares. The central slices are then either parallelograms, or hexagons with three pairs of opposite segments congruent and parallel.

     Just as we sliced the square with lines not through the origin, we can slice the cube will planes of the form ax + by + cz = e.  We will obtain either a vertex, or a triangle with three edges of different colors, or a quadrilateral with exactly on pair of parallel sides or a parallelogram or a pentagon with two pairs of parallel edges or a hexagon with three pairs of parallel edges.

Off-Center Slices with a = 1, b = 0, c = 0
1-0-0



Off-Center Slices with a = 1, b = 1, c = 0
1-1-0



Off-Center Slices with a = 1, b = 1, c = 2
1-1-2



Off-Center Slices with a = 1, b = 1, c = 1
1-1-1



SLICING HYPERCUBES BY HYPERPLANES IN FOUR-DIMENSIONAL SPACE

     These off-center slices of cubes will be the building blocks of central slices of the hypercube with vertices (+-1,+-1,+-1,+-1) by hyperplanes of the form ax + by + cz + dw = 0.  The hyperplane will meet the hyperplane w = 1 in a plane of the form ax + by + cz = -d, and this will slice the cube with w = 1 in one of the forms described above.  The intersection of the hyperplane with the hyperplane w = -1 will be congruent to the slice with the hyperplane w = 1, symmetrically situated with respect to the origin.  Similarly we will slice the cubes with z = 1 and z = -1 to find a pair of opposite vertices or opposite edges or congruent convex polygons with three, four, five, or six sides, symmetrically places with respect to the origin, and the same happens for the pair of cubes with y = +-1 or with x = +-1.  There will be at most eight of these intersections, fitting together to form a convex polyhedron in the 3-dimensional hyperplane.

     If all we want to see is the set of edges of this convex polyhedron, we can find the segments in its boundary by slicing a square with two coordinates fixed at 1 or -1 by the plane ax + by + cz + dw = 0. We will get 24 squares of this type since there are six choices for two of the four coordinates and, for each such pair of coordinate slots, two choices +-1 for the entry in that slot. Note that the final figure will be symmetric with respect to the origin, so any segment formed by slicing a square, say of the form (1,-1,x,w) will have a congruent opposite segment with points of the form (-1,1,-x,-w).

Rotation Giving Transition from Rectangular Prism Slice Hexagonal Prism Slice
rectangular_prism_to_hexagonal_prism

Rotation Giving Transition from Hexagonal Prism Slice to Octahedron Slice:
prism_to_octahedron




CONCLUSION

     In "Flatland: the Movie", the final sequence shows some views of projections of a hypercube but they do not correspond to the slices of the cube by Flatland that are the basis of the sign left by earlier visitors from the third dimension.  The images show perspective views of a hypercube being projected into three-dimensional space instead of symmetrical three-dimensional slices of a hypercube rotating about its center.  Just as Martin Gardner challenged his readers to find the central section of a cube with the largest area, Spherius, as well as the viewers of "Flatland: the Movie" will face the challenge of describing the slices by the hypercube in our three-dimensional universe and of figuring out which one of those hypercube slices has the greatest three-dimensional volume.  It is a problem that Martin Gardner will enjoy.