A NOTE TO THE READER: Because my Geometric Construction, a Niagara-class fast cruiser from Star Trek, is so large, I am sectioning it off into three parts: the body, the warp pods, and the background. Also, the compass makes an excellent measuring tool, so be prepared to use it as such. Final note: whenever I refer to something that hasn't been constructed yet or is being constructed, I am asking you to refer to your paper copy of my construction, which you can print out from my first post.
PRELIMINARY
PRELIMINARY
- Position the paper as though you were writing an essay, with the short sides at the top and bottom and the long sides on the left and right.
- Construct a line that divides the piece of paper in half lengthwise, by making a line parallel to the top of the page and bisecting it. Label it L1.
- Construct a line that divides the paper in half widthwise, by bisecting L1. Label the new line L2. Label the point at which the two lines intersect P1.
PART I: THE BODY
- Set compass width to the radius of the largest circle of the body. Construct a circle with center on L2 that touches the right edge of the paper. Label the center P2.
- Set compass width to the radius of the second-largest circle. Construct a circle with center on P2. Repeat for the third ring on the body.
- In the largest circle, inscribe a regular 16-sided figure. From each of the vertices, draw eight line segments that each divide the 16-sided figure in half and intersect P2.
- Set compass width to the distance between P2 and the small side of the larger coffin-shape. With the end of the compass on P2, make two tick marks on L2, one on either side of P2.
- Construct a perpendicular line to L2 that passes through the intersection of L2 and the tick mark created in step 4. Repeat for the other tick mark.
- Set compass width to the distance between the tick mark to the left of P2 and the endpoint of the coffin-shape's longer side (either one). Make two tick marks on the perpendicular line created in step 5, one on either side of the original tick mark.
- On the perpendicular line to the right of P2, make points where it intersects the inner circle. Draw two straight line segments connecting these points to the tick marks created in step 6.
- Set compass width to the distance between the tick mark created in step 6 (either one) and the leftmost tick mark created in step 4. Inscribe a circle with center on the leftmost tick mark created in step 4. The larger coffin-shape is done.
- Set compass width to the distance between P2 and the small side of the smaller coffin-shape. With the end of the compass on P2, make a tick mark on L2 to the left of P2.
- Construct a perpendicular line to L2 that passes through the tick mark created in step 9.
- Set compass width to the distance between L2 and the endpoint of the side (either one) of the smaller coffin-shape. With the end of the compass on the tick mark created in step 9, make two tick marks on the perpendicular line, one on either side of the tick mark created in step 9.
- Construct a perpendicular line to L2 that passes through P2.
- Set compass width to the distance between the endpoint of the long side (either one) of the smaller coffin-shape. With the end of the compass on P2, make two tick marks on the perpendicular line, one on either side of P2.
- Draw two straight line segments, one on either side of L2, that connect the tick marks created in step 11, and the tick marks constructed in step 13.
- Set compass width to the distance between the endpoints of the line segment created in step 14 (either one) and P2. Create a circle with center on P2. The smaller coffin-shape is done.
- Set compass width to the radius of the small circle inside the smaller coffin-shape. Create a circle with center on P2.
- Set compass width to the distance between P2 and the closer curved side of the serial number plaque. Create an arc, spanning at least 45 degrees on either side of L2. Repeat for the farther curved side of the serial number plaque.
- Set compass width to the distance between the intersection of L2 and the inner arc, and the vertex of the serial number plaque (either one). With the end of the compass on the intersection of L2 and the inner arc, make two tick marks on the inner arc, one on either side of the intersection.
- Draw two lines, each passing through P2, one of the tick marks made in step 18, and the outer arc. The serial number plaque is done
- The body is complete.
PART II: THE WARP PODS
- Construct a line parallel to L1 that passes through the point at which the outer circle of the body intersects L2 (label this point P3). Label this line L3.
- Set compass width to the distance between P3 and the endpoint of the larger parabola-shape (either one). With the end of the compass on P3, make two tick marks on L3, one on either side of P3.
- Set compass width to the distance between P3 and the vertex of the larger parabola-shape. With the end of the compass on P3, make a tick mark on L2.
- Set compass width to slightly less than half the width of the larger parabola-shape at its widest point (it's okay to estimate). Construct an oval with its narrow end (either one) on the tick mark made in step 3, and both narrow ends on L2. The oval should be to the right of the tick mark.
- Construct a line that bisects the oval widthwise. Where it intersects L2, make a point labeled P4.
- Construct two line segments, one on either side of L2. Each should connect a point where the line created in step 5 intersects the oval, and a tick mark created in step 2. Extend each of these until they intersect the outer circle of the body. Extend them the other way, also, so that the end-points are level with the vertex of the larger parabola-shape (it's okay to estimate). Label the top line L4 and the bottom line L5. The larger parabola-shape is done.
- Set compass width to the distance between the vertex of the larger parabola-shape and the vertex of the smaller parabola-shape. With the end of the compass on the vertex of the larger parabola-shape, make a tick mark on L2 to the left of the larger parabola-shape.
- Set compass width to the radius of the circle used to create the oval (either one). Construct a circle with center on L2 that intersects the tick mark made in step 7.
- Bisect the circle with a line that is perpendicular to L2. Draw two line segments, one on either side of L2, that begin at the intersection points of the line just created and the circle and are at right angles to the line just created. Extend these line segments so that they intersect with the larger parabola-shape. The smaller parabola-shape is done.
- Set compass width to the distance between the vertex of the larger parabola shape and the vertex of the smallest parabola-shape. With the end of the compass on the vertex of the larger parabola-shape, make a tick mark on L2 to the left of the larger parabola-shape.
- Set compass width to half the width of the smallest parabola-shape at its widest point. Construct an oval with its narrow end (either one) touching the tick mark made in step 10, and both narrow ends on L2. The oval should be to the right of the tick mark.
- Repeat steps 5 and 6 for the smallest parabola-shape. The smallest parabola-shape is done.
- Set compass width to the distance between the intersection of the outer circle of the body and L4, and the rightmost angle connected to L4. With the end of the compass on the intersection of the outer circle of the body and L4, make a tick mark on L4.
- At the tick mark created in step 13, construct a 60º angle facing up and to the left.
- Set compass width to the distance between the rightmost angle associated with L4 and the leftmost angle associated with L4. With the end of the compass on the vertex of the 60º angle created in step 14, make a tick mark on L4 to the left of the 60º angle.
- At the tick mark created in step 15, construct a 45º angle facing up and to the left.
- Repeat steps 13-16, but replacing "L4" with "L5", and replacing "up" with "down."
- Construct a line that is perpendicular to L2 and passes through the intersection of L2 and the vertex of the smaller parabola-shape. Label this line L6.
- Set compass width to the distance between the intersection of L2 and L6, and the vertex of the top warp pod. With the end of the compass on the intersection of L2 and L6, make two tick marks on L6, one on each side of the intersection.
- Repeat step 18 with the intersection of L2 and L3. Label this new line L7.
- Repeat step 19 with L7 instead of L6.
- Draw two line segments on either side of L2, connecting the tick marks made in step 19, and the tick marks made in step 21. Label the top one L8 and the bottom one L9.
- Set compass width to the radius of the circle used to make the smaller parabola-shape. Construct a circle to the right of L6 and with center on L8, that touches the intersection of L8 and L6.
- Bisect the circle created in step 23 with a line that is perpendicular to L8. Extend this line to cover the entire length of the page.
- Set compass width to slightly less than the radius of the smallest circle on the body (it's okay to estimate). Construct an oval to the left of L7, with narrow end on the intersection of L7 and L8, and both narrow ends on L8.
- Bisect the oval with a line that is parallel to L8.
- Draw two lines on both sides of L8, connecting the intersections of the circle and bisector from step 24, and the intersections of the oval and bisector from step 26. This will finish the sides of the top warp pod.
- Draw a line segment that is perpendicular to L8 and passes through the center of the leftmost circle used to construct the oval.
- Draw another line segment that is perpendicular to L8 and passes through the intersection of the leftmost circle used to construct the oval, and L8.
- Set compass width to the radius of the circle (either one) used to construct the oval. Draw a circle with the center at the point where the oval's bisector and L8 intersect. The oval's bisector will also bisect the circle.
- Draw two line segments, each one connecting one side of line segment created in step 28, and an intersection of the circle created in step 30 and its bisector.
- Set compass width to the distance between the intersection of the oval and L7, and the rightmost vertex of the equilateral triangle found on the top warp pod. With the end of the compass on the aforementioned intersection, make a tick mark on L8, to the left of L7.
- Construct an equilateral triangle that is bisected by L8, and has a vertex at the tick mark created in step 32.
- Set compass width to the distance between the bottom vertex of the triangle and the rightmost endpoint of the bottom line. With the end of the compass on the bottom vertex, make an arc that goes to the left of the equilateral triangle. Repeat with the end of the compass on the top vertex of the triangle.
- Set compass width to the distance between the bottom vertex of the triangle and the rightmost endpoint of the top line. With the end of the compass on the bottom vertex, make an arc that intersects the top arc made in step 34. Then, with the end of the compass on the top vertex, make an arc that intersects the bottom arc made in step 34. These two intersections will be the endpoints of the lines.
- Repeat steps 34 and 35 with the leftmost endpoints of the lines. When you are finished, you should have four points, two above and two below L8, that are the same distance apart.
- Draw two line segments connecting the top and bottom points created in steps 34-36. The top warp pod is finished.
- Repeat steps 23-37, replacing "L8" with "L9." Once you have done this, you should have a bottom warp pod that is identical to the top warp pod.
- The Warp Pods are done.
- The USS Princeton is done! Take a moment to admire your handiwork.
PART III: THE BACKGROUND
- Set compass width to the distance between P1 (the center of the page, if you've forgotten) and the right side of the page, at the point where it intersects L2. Construct a circle, with center on P1.
- Inscribe a regular hexagon inside that circle.
- Set compass width to the distance between P1, and the intersection of the hexagon and L2. Construct a circle, with center on P1.
- Inside the circle created in step 3, inscribe a regular octagon. Connect four of the points to make a square, and the other four points to make the same square rotated 45º.
- Set compass width to the length of one of the petals in one of the rosettes (any one). At three arbitrary points on the construction (one above the USS Princeton and to the right of L1, one below the USS Princeton and to the right of L1, and one below the USS Princeton and to the left of L1) (it's okay to estimate - see my construction for an idea), construct a circle.
- Inside one of the circles, inscribe a six-petalled rosette.
- Repeat step 6 for the other two circles.
- The Background is done.
- The construction is done.
SUBLIMINALLY
- Take a pen and go over the important lines - see my construction for which lines to go over.
- Congratulations! You just made a very complicated construction! Be proud of yourself! I'll bet you didn't know that you could do that with just a straight-edge and a compass.