Learning Math: Geometry
Investigate symmetry, one of the most important ideas in mathematics. Explore geometric notions of symmetry by creating designs and examining their properties. Investigate line symmetry and rotation symmetry; then learn about frieze patterns.
In This Session
Part A: Line Symmetry
Part B: Rotation Symmetry
Part C: Translation Symmetry and Frieze Patterns
Homework
Symmetry is one of the most important ideas in mathematics. There can be symmetry in an algebraic calculation, in a proof, or in a geometric design. It’s such a powerful idea that when it’s used in solving a problem, we say that we exploit the symmetry of the situation. In this session, you will explore geometric versions of symmetry by creating designs and examining their properties.
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For information on required and/or optional materials for this session, see Note 1.
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Learning Objectives
In this session, you will do the following:
- Learn about geometric symmetry
- Explore line or reflection symmetry
- Explore rotation symmetry
- Explore translation symmetry and frieze patterns
Key Terms
Previously Introduced
Coordinates: Points are geometric objects that have only location. To describe their location, we use coordinates. We begin with a standard reference frame (typically the x- and y-axes). The coordinates of a point describe where it is located with respect to this reference frame. They are given in the form (x,y) where the x represents how far the point is from 0 along the x-axis, and the y represents how far it is from 0 along the y-axis. The form (x,y) is a standard convention that allows everyone to mean the same thing when they reference any point.
Reflection: Reflection is a rigid motion, meaning an object changes its position but not its size or shape. In a reflection, you create a mirror image of the object. There is a particular line that acts like the mirror. In reflection, the object changes its orientation (top and bottom, left and right). Depending on the location of the mirror line, the object may also change location.
Rotation: Rotation is a rigid motion, meaning an object changes its position but not its size or shape. In a rotation, an object is turned about a “center” point, through a particular angle. (Note that the “center” of rotation is not necessarily the “center” of the object or even a point on the object.) In a rotation, the object changes its orientation (top and bottom). Depending on the location of the center of rotation, the object may also change location.
Translation: Translation is a rigid motion, meaning an object changes its position but not its size or shape. In a translation, an object is moved in a given direction for a particular distance. A translation is therefore usually described by a vector, pointing in the direction of movement and with the appropriate length. In translation, the object changes its location, but not its orientation (top and bottom, left and right).
New in This Session
Frieze Pattern: A frieze pattern is an infinite strip containing a symmetric pattern.
Glide Reflection: A glide reflection is a combination of two transformations: a reflection over a line followed by a translation in the same direction as the line.
Reflection or Line Symmetry: A polygon has line symmetry, or reflection symmetry, if you can fold it in half along a line so that the two halves match exactly. The folding line is called the line of symmetry.
Rotation Symmetry: A figure has rotation symmetry if you can rotate (or turn) that figure around a center point by fewer than 360° and the figure appears unchanged.
Symmetry: A design has symmetry if you can move the entire design by either rotation, reflection, or translation, and the design appears unchanged.
Translation Symmetry: Translation symmetry can be found only on an infinite strip. For translation symmetry, you can slide the whole strip some distance, and the pattern will land back on itself.
Vector: A vector can be used to describe a translation. It is drawn as an arrow. The arrowhead points in the direction of the translation, and the length of the vector tells you the length of the translation.
Notes
Note 1
Materials Needed:
- Mira (a transparent image reflector)
- tracing paper or patty paper and fasteners
- large paper or poster board (optional)
- scissors
- markers
Mira
You can purchase a Mira from the following source:
ETA/Cuisenaire
500 Greenview Court
Vernon Hills, IL 60061
800-445-5985/847-816-5050
800-875-9643/847-816-5066 (fax)
http://www.etacuisenaire.com/
Sections
Sessions
<span>This article is a follow-up on my last one, entitled Rodin Fibonacci Wheel Symmetries.
The first key to taking the wheel to the next dimension is the six hexagrams, and the triangles they contain.We will always count the numbers on a triangle clockwise in order to label them. Think of this as the experience of time in one direction. It is helpful, however to think of a backwards stream of information through time. Thus we have 3-6-9; 6-3-9; 1-1-1; 8-8-8; 1-4-7; 5-2-8; 1-7-4; 2-5-8.</span>
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<a href='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTSpF1a7M9I/AAAAAAAAABU/ssvKK_Psfv0/s1600/Rodin+Fibonacci+Symmetries.png' imageanchor='1'><img border='0' height='296' n4='true' src='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTSpF1a7M9I/AAAAAAAAABU/ssvKK_Psfv0/s320/Rodin+Fibonacci+Symmetries.png' width='320' /></a></div>
The next key I discovered is the application of the Buckminster Fuller Vector Flexor 'jitterbug'. This is a beautiful geometric dance which Bucky believed was a model for the Universe. It is significant I think, because it shows the crucial transformation from cubic/octahedral symmetry to dodecahedral/icosahedral symmetry. Take a look at these animations to get an idea of how this works.
<a href='http://www.youtube.com/watch?v=HefLC3PW8XQ&feature=related'>http://www.youtube.com/watch?v=HefLC3PW8XQ&feature=related</a>
<a href='http://www.youtube.com/watch?v=FfViCWntbDQ&feature=related'>http://www.youtube.com/watch?v=FfViCWntbDQ&feature=related</a>
Now how does this apply to the Rodin Fibonacci Wheel? Watch closely.
Our starting point will always be the cubeoctahedron, which is known in physics as vector equilibrium.
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<a href='http://4.bp.blogspot.com/_XEdakZFo-OQ/TTSp3FTSUqI/AAAAAAAAABY/_LwXYTNr5uI/s1600/Cubeocta.bmp' imageanchor='1'><img border='0' n4='true' src='http://4.bp.blogspot.com/_XEdakZFo-OQ/TTSp3FTSUqI/AAAAAAAAABY/_LwXYTNr5uI/s1600/Cubeocta.bmp' /></a></div>
This will correspond to the first two triangles, 3-6-9 and 6-3-9. This also corresponds to a noble gas in Walter Russel's system (diagram in my post Template for Universal Mathematics): asexuality, maximum inertia, maximum stablitiy, maximum complexity/softness of crystallization.
The cubeoctahedron then can tilt to the right or left, where it becomes an icosahedron at a tilt of 15 degrees. This is the same tilt angle as the four askew triangles, 1-4-7; 5-2-8; 7-4-1 and 2-5-8!!! When the icosahedron is closing into the octahedron, it is male, and when it opens back into the cubeoctahedron it is female.
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<a href='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTStUYX9_UI/AAAAAAAAABk/pkl3RbnBTLc/s1600/Jitterbug+Icosa+flipped.jpg' imageanchor='1'><img border='0' n4='true' src='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTStUYX9_UI/AAAAAAAAABk/pkl3RbnBTLc/s1600/Jitterbug+Icosa+flipped.jpg' /></a><a href='http://1.bp.blogspot.com/_XEdakZFo-OQ/TTStFcCdA_I/AAAAAAAAABg/-Jknuk8qgH0/s1600/Jitterbug+Icosa+flipped.jpg' imageanchor='1'><img border='0' n4='true' src='http://1.bp.blogspot.com/_XEdakZFo-OQ/TTStFcCdA_I/AAAAAAAAABg/-Jknuk8qgH0/s1600/Jitterbug+Icosa+flipped.jpg' /></a></div>
At the maximum point of the wave, the figure collapses to an octahedron. This corresponds with carbon: bisexuality, maximum compression, maximum resistance, simplicity and hardness of crystallization, and the appearance, due to motion, of the stability of form.
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<a href='http://2.bp.blogspot.com/_XEdakZFo-OQ/TTUDcf-BDYI/AAAAAAAAACI/Gv6_KEzkbxk/s1600/Octahedron.jpg' imageanchor='1'><img border='0' n4='true' src='http://2.bp.blogspot.com/_XEdakZFo-OQ/TTUDcf-BDYI/AAAAAAAAACI/Gv6_KEzkbxk/s1600/Octahedron.jpg' /></a></div>Despite the 3-dimensional transformation between these three shapes, a 2-dimensional hexagram is visible, when we center our perspective on a triangular face. So we see a motion which moves in a cycle, but can change direction at four turning points: 9-6-3; 3-6-9; 1-1-1; and 8-8-8. In this way, it can be steered into infinitely complex forms. We can look at it like a flow chart indexed by the angle of the triangle that is front and centered, corresponding with the 8 triangles of our 24 number circle.
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<a href='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTUD94MeMZI/AAAAAAAAACM/dEzOAxAn4Mw/s1600/Rodin+Fibonacci+Jitterbug+Flow+Chart.png' imageanchor='1'><img border='0' height='320' n4='true' src='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTUD94MeMZI/AAAAAAAAACM/dEzOAxAn4Mw/s320/Rodin+Fibonacci+Jitterbug+Flow+Chart.png' width='286' /></a></div>
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Now compare this with the eight trigrams around a circle which form the basis of the I Ching.
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<a href='http://1.bp.blogspot.com/_XEdakZFo-OQ/TTSymrjzlwI/AAAAAAAAAB0/o4yZ1uskXTY/s1600/I+Ching+Trigram+Names.jpg' imageanchor='1'><img border='0' n4='true' src='http://1.bp.blogspot.com/_XEdakZFo-OQ/TTSymrjzlwI/AAAAAAAAAB0/o4yZ1uskXTY/s1600/I+Ching+Trigram+Names.jpg' /></a></div>
The four trigrams at the top/bottom/left/right are in positions of relative balance, at the zero points and peaks of the wave, whereas the four trigrams at diagonals are in the transitional phase. Because the left side of the circle is a mirrored inverse of the right side, we can unite the opposing hexagrams for a deeper understanding of this system. The noble gas is both Creative AND Receptive, it contains both the memories of every action, and the imagination which will create the future. The dense matter of maximum potential at the peak of the wave is Radiant AND Dark. That is, hot and cold, light and dark, dense motion at the center and tenuous space surrounding it are in maximum opposition. We can simplify and bring these two together into a kind of magic square/circle.
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<a href='http://1.bp.blogspot.com/_XEdakZFo-OQ/TTS5Gw1my7I/AAAAAAAAAB8/0KaGR3wIjMw/s1600/I+Ching+Magic+Circle.png' imageanchor='1'><img border='0' height='320' n4='true' src='http://1.bp.blogspot.com/_XEdakZFo-OQ/TTS5Gw1my7I/AAAAAAAAAB8/0KaGR3wIjMw/s320/I+Ching+Magic+Circle.png' width='308' /></a></div>
3-6-9 triangle becomes 3 according to multiples of 3; likewise 6-3-9 becomes 6. 1-1-1 becomes 1, 8-8-8 becomes 8. 1-4-7 becomes 4 because it is powers of 4, and 7-4-1 is powers of 7. 2-5-8 and 5-2-8 are abbreviations of halving and doubling cycles, or 5 and 2, respectively.
Symmetry Lending
Now imagine that you can not only change direction from clockwise or counter-clockwise, but that, at any stage of the cycle, you can choose a new center triangle to start from. This gives us a model of the 4-dimensional 24-cell, which can nest all five platonic solids. Look at this view where the octahedron goes from a plane, to fully formed, back to a plane again.Axis Of Symmetry
<div><a href='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTSzJQcAmXI/AAAAAAAAAB4/ZuTzfJnMTjw/s1600/24-Cell+in+Cubeoctahedron+Envelope.gif' imageanchor='1'><img border='0' height='160' n4='true' src='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTSzJQcAmXI/AAAAAAAAAB4/ZuTzfJnMTjw/s320/24-Cell+in+Cubeoctahedron+Envelope.gif' width='320' /></a></div>
To view it as a 3-d hologram, cross your eyes and try to bring the two images into resolution together, a lot like a Magic Eye. I recommend printing the image if you are doing this a lot, as staring at a computer screen with your eyes crossed will give you a headache. From this page on Hyperspace. <a href='http://home.comcast.net/~eswab/hyprspac.html'>http://home.comcast.net/~eswab/hyprspac.html</a>
Line Of Symmetry
Furthermore, it unfolds the pattern to 64 total permutations, just like the I Ching. If this can be shown to be a workable system, it would throw a completely new spin (literally) on the I Ching; there are various states of motion described by the different hexagrams. From this perspective, we could use Nassim Haramein's 64 tetrahedron vector-matrix grid ( <a href='http://www.theresonanceproject.org/graphics.html'>http://www.theresonanceproject.org/graphics.html</a> ) as a starting point. I am trying to find someone who could model this matrix going through the Vector Flexor Jitterbug Dance.<div>
<a href='http://1.bp.blogspot.com/_XEdakZFo-OQ/TTS59KLD_sI/AAAAAAAAACE/EB83dKb3tF8/s1600/Isotropic+Vector+Matrix.jpg' imageanchor='1'><img border='0' height='150' n4='true' src='http://1.bp.blogspot.com/_XEdakZFo-OQ/TTS59KLD_sI/AAAAAAAAACE/EB83dKb3tF8/s320/Isotropic+Vector+Matrix.jpg' width='320' /></a><a href='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTS541LB8dI/AAAAAAAAACA/TFd25lJopOQ/s1600/64-Tetrahedron+Grid.jpg' imageanchor='1'><img border='0' n4='true' src='http://3.bp.blogspot.com/_XEdakZFo-OQ/TTS541LB8dI/AAAAAAAAACA/TFd25lJopOQ/s1600/64-Tetrahedron+Grid.jpg' /></a></div>
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Symmetry Furniture
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More to come, feedback is encouraged/appreciated. Peace to All.