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Abstract Algebra Presentation on Geometric Constructions: Looking at constructible numbers, roots of polynomials, and fields. This information can be extended to complete impossibility proofs.
My presentation on geometric constructions for my abstract algebra class! Using fields, I’ll show you why you can construct certain numbers, why you can’t construct certain numbers, how this relates to roots of polynomials.
This is a picture of 100 of solutions.
In each, a quarter of the cross is coloured in red. You can explore a prezi of them here. A high-resolution picture can be downloaded here (PNG 4MB) and the original SVG file here.
Ideas used to construct my solutions by David Butler.
Fraction addition and subtraction.
Halving retangles and triangles using symmetry.
Adding a bit to a shape and taking the same bit don’t change the area.
Moving the apex of a triangle parallel to its base doesn’t change the area.
Shearing a shape doesn’t change its area.
Making a shape half as high and twice as wide doesn’t change the area.
Shrinking all the lengths by half makes a shape with a quarter the area.
The formula for the area of a trapezium.
The formula for the area of a crossed trapezium.
The Lune of Hippocrates.
Pythagoras’ theorem.
Rotational symmetry.
The sum of a geometric series.
Stereographic projection of regular polychora (4D polytopes) by emileokada.
“Having made a demonstration that successfully manages to project regular 3D polytopes I decided it would be worth a shot trying to generalize to higher dimensions. Doing some minor computations shows that generalizing to higher dimensions is in fact trivial! ”
Some major points to take note of are:
(1) You want to “explode” the polytope onto the n-sphere. Take a point in n-space given by the position vector ,
then it’s image on the n-sphere is given by
(2) You want to project a point on the n-sphere into a space tangent to the n-sphere i.e. a subspace of n-1 dimensions. Thus the stereographic projection of a point in n-space to a point in n-1 space is given by
where
is a point on the n-sphere. Taking these facts into account it was easy to modify the code to produce the following animations of the simplex, hypercube, 16-cell and 24-cell:
GIF 1: simplex.gif
GIF 2: tesseract.gif
GIF 3: hexadecachoron.gif
GIF 4: Icositetrachoron.gif
The 120-cell and the 600-cell as they took way to long to render.
Projections of the world onto convex polyhedra.
In the chapter Courious Maps of M. Gardner's Time Travel (W.H. Freeman and Company, New York, 1988), many world's maps are presented, some of them use a projection on a convex solid, that is unfolded into a plane net.
The mathematician Charles Sanders Peirce designed a conformal map, namely a projection on eight isosceles right triangles that may be regarded as the faces of an octahedron, flattened until a space diagonal is zero. B.J.S. Cahill patented his butterfly map in 1913. The world is projected onto a regular octahedron. R. Buckminster Fuller's first Dymaxion map was a projection of the world onto the fourteen faces of a cuboctahedron. In March 1, 1943 the map was completed by staff artists of Life. At about the same time, Irving Fisher designed Likaglobe, which folds into an icosahedron. In 1954 Fuller copyrighted his Dymaxion Skyocean Projection World Map, which slightly differed from Fisher's Likaglobe.
The following animations use LiveGraphics3D animation applet. To invoke an animation click on a figure. Then drag to rotate, press Shift and drag vertical to change size, or horizontal to rotate. Press S to get stereo view. See more at here. [mi.sanu.ac.rs]
The next figure shows a cylindrical projection with a maze on it. The task is to join the black and the grey dot on it or for instance to find a path from Belgrade to New York. A set of 19 figures of this sort can be downloaded from here.
Belt-extruder
Putting the extruder stepper on a fixed position to minimize moving weight by Ulrich Baer.
Independent belts for X/Y/E. The X/Y using double precision (Gun tackle). He isn't using a single belt like the H-BOT design, to minimize interaction between axis. Further All Motors (Stepper or Servo) are outside and fixed, so size and weight doesn't matter (well inertia of the armature does). The Z- Axis (not shown) will be 3(or 4) Spindles so calibration of the bed (tilting) is possible (maybe also 5-axis printing). He still think about an upside down printing process with a cooled teflon plattform around the nozzle. Using belts allow to scale this (maybe using Vive Lighthouses for precision at room scale) - as pulleys can be printed and additonal Belts are inexpensive this should keep costs still low. Only the Extruder belt is looped and may be a problem on bigger scales as this would be a non standart part.
Geometric Constructions, watercolor
Me at age 17: “This math book sucks! I’ll never use this geometry crap in real life!”
Me at age 35: *hovering over a piece of felt with a compass, ruler, and pencil, trying to make the best pentagon ever so my kindergarten class can learn about shapes with cool felt visual aids*
Nine-point circle
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:
The midpoint of each side of the triangle
The foot of each altitude
The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes).
The nine-point circle is also known as Feuerbach's circle, Euler's circle, Terquem's circle, the six-points circle, the twelve-points circle, the n-point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the nine-point center of the triangle
Gif (2): Even if the orthocenter and circumcenter fall outside of the triangle, the construction still works.
Image (3): The nine point circle and the 16 tangent circles of the orthocentric system.
