When we say "dodecahedron" we often mean "regular dodecahedron" (in other words all faces are the same size and shape), but it doesn'thaveto be - this is also a dodecahedron, even though all faces are not the same.
Background: You can construct a dodecahedron by cutting 12 identical, regular pentagon faces, beveling all the edges of these to half the dihedral angle and gluing them all together. The joints become the edges. Where 3 faces meet, you have the vertex. Simple enough.Now let's say you want to build a dodecahedron, but instead of boards for the faces, you want the boards to be the edges. Three edges come together at each vertex, etc.
Relative to some dimension of the paper itself, how would you determine/describe the two planes you'd cut at the end of each poster board in order to allow the three poster boards to meet and be correctly oriented in three dimensions? Let's say you want one surface of the board to split the dihedral. (i.e. boards are halfway between being coplanar with the 2 faces on either side.)
Tessellations, or regular divisions of the plane, are arrangements of closed shapes that completely cover the plane without overlapping and without leaving gaps. For shapes to fill the plane without overlaps or gaps, their angles, when arranged around a point, must have measures that add up to exactly 360 degrees.
Typically, the shapes making up a tessellation are polygons or
similar regular shapes (like square tiles used on floors). Escher exploited these basic patterns in his tessellations, applying reflections, translations, and rotations to obtain a greater variety of patterns. He also “distorted” these shapes to form animals, birds, and other figures. These distortions had to obey the three, four, or six- fold symmetry of the underlying pattern in order to preserve the tessellation.
Over 2,200 years ago, ancient Greeks were decorating their homes with tessellations, making elaborate mosaics from tiny, square tiles. Early Persian and Islamic artists also created spectacular tessellating designs. More recently, the Dutch artist M. C. Escher used tessellation to create enchanting patterns of interlocking creatures, such as birds and fish.
Carver High art students have taken Escher's tessellations to the
next level by using their names as organic or geometric shapes. Through careful guidance my students have learned how to create graffiti styled fonts and fit the design within triangular shapes that join to make a beautiful geometric design.
As a recurring project, Carver art students will create 3 dimensional graphic organizers that will simultaneously illustrate the use of vocabulary words and definitions while increasing reading fluency skills. A graphic organizer is a visual and graphic display that depicts the relationships between facts, terms, and or ideas within a learning task.
Graphic organizers are also sometimes referred to as knowledge maps, concept maps, story maps, cognitive organizers, advance organizers, or concept diagrams. The 3-D graphic organizer will be one of many different recurring assignments that students will compile to create a working portfolio. Portfolio design should provide students with the opportunities to become more reflective about their own work, while demonstrating their abilities to learn and achieve in academics.
Graphic organizers have been applied across a range of curriculum subject areas. Although reading is by far the most well studied application, science, social studies, language arts, and math are additional content areas that are represented in the research base on graphic organizers. Operations such as mapping cause and effect, note taking, comparing and contrasting concepts, organizing problems and solutions, and relating information to main ideas or themes can be beneficial to many subject areas. The observed benefits in these subject areas go beyond those known to occur in reading comprehension (Bulgren, Schumaker, & Deshler, 1988; Darch, Carnine, & Kammenui, 1986; Herl, O'Neil, Chung, & Schacter, 1999; Willerman & Mac Harg, 1991).
Moore and Readence's (1984) meta-analysis suggests that gains in vocabulary knowledge following graphic organizer use may be even greater than gains in comprehension. The average effect size for the 23 studies reviewed was more than twice as large as that reported for comprehension. Thus, graphic organizers appear to be a very effective tool for improving vocabulary knowledge.
References
Bulgren, J., Schumaker, J. B., & Deschler, D. D. (1988). Effectiveness of a concept teaching routine in enhancing the performance of LD students in secondary-level mainstream classes. Learning Disability Quarterly, 11(1), 3-17.
Darch, C. B., Carnine, D. W., & Kammeenui, E. J. (1986). The role of graphic organizers and social structure in content area instruction. Journal of Reading Behavior, 18(4), 275-295.
Herl, H. E., O'Neil, H. F. Jr., Chung, G. K. W. K. & Schacter, J. (1999). Reliability and validity of a computer-based knowledge mapping system to measure content understanding. Computers in Human Behavior, 15(3-4), 315-333.
Moore, D. W., & Readence, J. E. (1984). A quantitative and qualitative review of graphic organizer research. Journal of Educational Research, 78(1), 11-17.
Willerman, M., & Mac Harg, R. A. (1991). The concept map as an advance organizer. Journal of Research in Science Teaching, 28(8), 705-712.
This course is designed for beginner artists who have an interest in learning how to draw and sketch as well as paint and build 3 dimensional works of art-- or simply want to improve their skills. It covers the tools you'll need to draw, paint, design and build as well as provide plenty of exercises to practice using different techniques and styles. It will cover drawing people, places, things, and even abstract concepts. For best results, I highly recommend that students take their time going through this course.
This introductory Course also emphasizes the analysis and exploration of basic drawing techniques in the visual interpretation of natural and fabricated forms. Students will be introduced to a variety of drawing media. Two hours lecture, four hours studio each week. The fundamental objective of the course will be to develop an understanding of drawing as a two dimensional visual equivalent of the three dimensional world. Students will thereby have a more solid foundation from which to transfer their two-dimensional skills to a 3 dimensional environment. The emphasis will be on creative problem solving, through the development of a "language" of drawing, designing and building.
This course also involves the observation and translation of three-dimensional form into two-dimensional drawings. Starting with simple shapes and progressing to more complex organic forms, students will build skill levels in composition, line quality, use of tone and human anatomy. As seen below, students are learning the basic concepts of creating a 3 dimensional object on a 2 dimensional surface:
Upon successful completion of this phase of the course students should be able to: Articulate and solve basic problems of form in space through line and mass.
•Learn to layout a drawing: use sighting techniques to create object relationships on paper which relate convincingly to what is being observed.
Students will begin to control the direction of observed contour line, and to control line
weight as it relates to contour edge and outline.
•Students will be able to use the elements and principles of design to compose a drawing based on observation.
Students will be able to develop ideas through the thumbnail drawing process.
•Students will be able to analyze drawings, and using acquired vocabulary, identify
opportunities for improvement.
•Students will become aware of the art historical context of drawing.
Generate drawings that display differences in lighting and value.
•Become acquainted with and learn to use basic drawing tools: pencils of varying hardness’s, charcoal and papers.
•Learn to “see” value intervals and translate what is seen to the two-dimensional drawing surface.
Generate drawings that utilize economy of line
•Students will acquire the habit of keeping a sketchbook to record drawn observations, thumbnail ideas, and notes.
