[size=100][br]The image in the applet is based on the work [i]“Variation 1”[/i], which is a part of the series [i]“15 Variations on a Single Theme”[/i] designed by the Swiss designer Max Bill in 1938. [/size]
GeoGebra – a place for diverse applications and explorations
This presentation will demonstrate a number of applets showing the diversity of ways in which we could use GeoGebra - from demonstrating well known mathematical facts to discovering new patterns; for modeling natural phenomena or bringing together mathematics and the arts.
Table of Contents
- Visualization
- Conic Sections
- Möbius strip
- Discover Patterns and Generalize
- From Pascal's triangle to Sierpinski's triangle
- Patterns in Pascal’s Triangle Modulo m
- Patterns in Nature
- Gnomonic Growth of the Nautilus
- Measuring Equiangular Spirals in Nature
- Sunflowers and Special Numbers
- Take an Unusual Path
- Regular Pentagon and the Golden Ratio
- Math Drills
- Exact Values of Some Trig Functions
- Asymptotes of Rational Functions
- Fun with colors
- Art with Regular Polygons. Coloring.
Conic Sections
A conic section is the intersection of a plane and a cone. The three conic sections ellipse, parabola or hyperbola can be produced by changing the slope of the plane (that is, the angle between the axes of the cone and the intersecting plane).[br]In the applet below we consider an infinite cone with an angle [math]\alpha[/math].[br][*]Play the animation to see a demonstration of the conics, hyperbola, parabola, ellipse and circle.[br][*]Stop the animation and explore additional special cases by changing the angle of the plane (drag the orange point) and the location of the plane (drag the brown point).
From Pascal's triangle to Sierpinski's triangle
[i]Introducing Fractals[/i], by Nigel Lesmoir-Gordon, Will Rood & Ralph Edney
Gnomonic Growth of the Nautilus
“The image below is a cross section of the chambered nautilus. The chambered nautilus builds its shell in stages, each time adding another chamber to the already existing shell. At every stage of its growth, the shape of the chambered nautilus shell remains the same–the beautiful and distinctive spiral.[br]This is a classic example of gnomonic growth–each new chamber added to the shell is a gnomon of the entire shell. The gnomonic growth of the shell proceeds, in essence, as follows: Starting with its initial shell (a tiny spiral similar in all respects to the adult spiral shape), the animal builds a chamber (by producing a special secretion around its body that calcifies and hardens). The resulting, slightly enlarged spiral shell is similar to the original one. Each new chamber adds a gnomon to the shell, so the shell grows and yet remains similar to itself.”[br]From [i]Excursions In Modern Mathematics[/i], by Tannenbaum
Regular Pentagon and the Golden Ratio
A [i]golden triangle[/i] is an isosceles triangle in which the ratio of the two congruent sides to the third side is the golden ratio [math]\approx1.618...[/math]. The Golden triangle has angles [math]\text{72^\circ-72^\circ-36^\circ.}[/math][br]Use the information about the sides and the angles of the five congruent isosceles trapezoids to find all pairs of segments in the regular pentagon that are in golden ratio.[br][list][*]Drag the slider enough to fold one trapezoid. [br][/*][*]Find all golden triangles in the regular pentagon ABCDE formed by the sides and the creases.[br][/*][*]List all pairs of segments in golden ratio.[/*][/list]Repeat the three steps above for the remaining trapezoids.[br][br]
Exact Values of Some Trig Functions
Exact Values of Some Trig Functions
Art with Regular Polygons. Coloring.
Use the basic color wheel on the right and the color mixers below it to color Max Bill's design.[br][b]To apply a color on the design:[/b][br][list=1][*]Click on a desired color from the color wheel to pick up the color.[br][/*][*]Click on the area of the design you want to color.[br][/*][*]Click again on the selected color on the color wheel to deselect it.[/*][/list][b]To modify a color on the color wheel:[/b][br][list=1][*]Click on a triangle from the color wheel.[br][/*][*]Drag the sliders to modify the color of the selected triangle.[br][/*][*]Click again on the same triangle to set the color.[/*][/list]