Duality

What is it?

$DualsPlatonicSolids_1000.gif$
Image from MathWorld

The dual of a convex polyhedron is obtained by placing a vertex of the new (dual) polyhedron in the center of each face of the original polyhedron and if two faces of the original polyhedron are joined by an edge, then the vertices in the centers of those faces are joined by an edge in the dual. Basically, the vertices of one polyhedron correspond to the faces of the other. The dual of the dual is the original polyhedron. A point in the center of each of the 6 faces of a cube connected as described results in an octahedron.

A characteristic of dual polyhedra is that the dual of the dual is the original polyhedron. The dual of the octahedron is the cube. By the construction of the dual each face in the original corresponds to a vertex in the dual (the vertex placed in the center of the face), and each vertex in the original corresponds to a face in the dual. The number of edges in a polyhedron and its dual are always equal, since a new edge is constructed for every edge connecting two faces in the original, so the counts for the dual are the same as the counts in the original, but with the number of faces and vertices interchanged.

The dual of the icosahedron, {3, 5}, is the dodecahedron, {5, 3}, and vice versa. The twenty 3-sided faces and twelve 5-way corners of the icosahedron correspond to the twenty 3-way corners and twelve 5-sided faces of the dodecahedron. Each has thirty edges. The compound of the icosahedron and dodecahedron shows these relationships very clearly.
$icos-dodec.gif$
Image from /www.georgehart.com

For platonic solids the dual to {p, q} is always {q, p}. When one takes the duals of the Archimedean solids however, one gets a new class of solids, the Archimedean duals.

member pmoews created an openSCAD file for Duals of Polyhedra.

If the points are the vertices of an Archimedean solid the output is a Catalan solid.

If the points are a solution to the Thomson problem, the output is the convex hull dual.

If the points are the centers of the faces of a polyhedron all equidistant from the origin, the output is the polyhedron.

More than one set of points can be used as input so as to output compound polyhedra. For example the Archimedean polyhedra cannot be output directly as their faces are not equidistant from the origin. However the Archimedean solids can be generated by treating each set of faces separately and combining them.

Some sample scad files and their outputs are provided. They are:

A program to generate a cuboctahedron from its face centers:
cuboctahedron_to_self.scad

A program to generate a rhombic dodecahedron from the vertices of the cuboctahedron:
cuboctahedron_to_dual.scad

A program to generate the cuboctahedron - rhombic dodecahedron compound:
compound_cuboctahedron_dual.scad

A program to generate the material common to a cuboctahedron and a rhombic dodecahedron:
common_cuboctahedron_dual.scad

1.1	Describe objects, imaginary or real, that might be modeled or made differently and suggest ways in which the objects can be changed, fixed, or improved	√
1.2	Investigate prior solutions and ideas from books, magazines, family, friends, neighbors, and community members	√
1.3	Generate ideas for possible solutions, individually and through group activity; apply age-appropriate mathematics and science skills; evaluate the ideas and determine the best solution; and explain reasons for the choices	√
1.4	Plan and build, under supervision, a model of the solution using familiar materials, processes, and hand tools	√
1.5	Discuss how best to test the solution; perform the test under teacher supervision; record and portray results through numerical and graphic means; discuss orally why things worked or didn't work; and summarize results in writing, suggesting ways to make the solution better	√

T1.1	Identify needs and opportunities for technical solutions from an investigation of situations of general or social interest.	√
T1.1a	Identify a scientific or human need that is subject to a technological solution which applies scientific principles	√
T1.2	Locate and utilize a range of printed, electronic, and human information resources to obtain ideas.	√
T1.2a	Use all available information systems for a preliminary search that addresses the need.	√
T1.3	Consider constraints and generate several ideas for alternative solutions, using group and individual ideation techniques (group discussion, brainstorming, forced connections, role play); defer judgment until a number of ideas have been generated; evaluate (critique) ideas; and explain why the chosen solution is optimal.	√
T1.3a	Generate ideas for alternative solutions	√
T1.3b	Evaluate alternatives based on the constraints of design	√
T1.4	Develop plans, including drawings with measurements and details of construction, and construct a model of the solution, exhibiting a degree of craftsmanship.	√
T1.4a	Design and construct a model of the product or process	√
T1.4b	Construct a model of the product or process	√
T1.5	In a group setting, test their solution against design specifications, present and evaluate results, describe how the solution might have been modified for different or better results, and discuss trade-offs that might have to be made.	√
T1.5a	Test a design	√
T1.5b	Evaluate a design	√

1.1	Initiate and carry out a thorough investigation of an unfamiliar situation and identify needs and opportunities for technological invention or innovation	√
1.2	identify, locate, and use a wide range of information resources including subject experts, library references, magazines, videotapes, films, electronic data bases and online services, and discuss and document through notes and sketches how findings relate to the problem
1.3	generate creative solution ideas, break ideas into the significant functional elements, and explore possible refinements; predict possible outcomes using mathematical and functional modeling techniques; choose the optimal solution to the problem, clearly documenting ideas against design criteria and constraints; and explain how human values, economics, ergonomics, and environmental considerations have influenced the solution	√
1.4	develop work schedules and plans which include optimal use and cost of materials, processes, time, and expertise; construct a model of the solution, incorporating developmental modifications while working to a high degree of quality (craftsmanship)	√
1.5	in a group setting, devise a test of the solution relative to the design criteria and perform the test; record, portray, and logically evaluate performance test results through quanitative, graphic, and verbal means; and use a variety of creative verbal and graphic techniques effectively and persuasively to present conclusions, predict impacts and new problems, and suggest and pursue modifications

1.1	Use computer technology,traditional paper-based resources,and interpersonal discussions to learn, do, and share science in the classroom	√
1.2	Select appropriate hardware and software that aids in wordprocessing, creating databases, telecommunications, graphing, data display, and other tasks	√
1.3	Use information technology to link the classroom to world events

1.1	Use a range of equipment and software to integrate several forms of information in order to create good-quality audio, video, graphic, and text-based presentations.
1.2	Use spreadsheets and database software to collect, process, display, and analyze information. Students access needed information from electronic databases and on-line telecommunication services.
1.3	Systematically obtain accurate and relevant information pertaining to a particular topic from a range of sources, including local and national media, libraries, muse- ums, governmental agencies, industries, and individuals.
1.4	Collect data from probes to measure events and phenomena.
1.4a	Collect the data, using the appropriate, available tool
1.4b	Organize the data
1.4c	Use the collected data to communicate a scientific concept	√
1.5	Use simple modeling programs to make predictions.

Duality

What is it?

Challenge

Standards

Mathematics, Science, and Technology

CDOS

Career Development and Occupational Studies

Essential Questions

Extruding

TinkerCad

Blender

OmNomNom Creator

Inkscape to OpenSCAD converter

1.1	Understand and use the more advanced features of word processing, spreadsheets, and database software.
1.2	Prepare multimedia presentations demonstrating a clear sense of audience and purpose. (Note: Multimedia may include posters, slides, images, presentation software, etc.)	√
1.2a	Extend knowledge of physical phenomena through independent investigation, e.g., literature review, electronic resources, library research
1.2b	Use appropriate technology to gather experimental data, develop models,and present results	√
1.3	Access, select, collate, and analyze information obtained from a wide range of sources such as research databases, foundations, organizations, national libraries, and electronic communication networks, including the Internet.
1.3a	Use knowledge of physics to evaluate articles in the popular press on contemporary scientific topics
1.4	Utilize electronic networks to share information.	√
1.5	Model solutions to a range of problems in mathematics, science, and technology, using computer simulation software.	√
1.5a	Use software to model and extend classroom and laboratory experiences,recognizing the differences between the model used for understanding and real-world behavior	√

2.1	Explore, use, and process a variety of materials and energy sources to design and construct things.	√
2.2	Understand the importance of safety, cost, ease of use, and availability in selecting tools and resources for a specific purpose.
2.3	Develop basic skill in the use of hand tools
2.4	Use simple manufacturing processes (e.g., assembly, multiple stages of production, quality control) to produce a product	√
2.5	Use appropriate graphic and electronic tools and techniques to process information.	√

2.1	Choose and use resources for a particular purpose based upon an analysis and understanding of their properties, costs, availability, and environmental impact	√
2.2	Use a variety of hand tools and machines to change materials into new forms through forming, separating, and combining processes, and processes which cause internal change to occur	√
2.3	Combine manufacturing processes with other technological processes to produce, market, and distribute a product
2.4	Process energy into other forms and information into more meaningful information.

2.1	Test, use, and describe the attributes of a range of material (including synthetic and composite materials), information, and energy resources	√
2.2	Select appropriate tools, instruments, and equipment and use them correctly to process materials, energy, and information	√
2.3	Explain tradeoffs made in selecting alternative resources in terms of safety, cost, properties, availability, ease of processing, and disposability
2.4	Describe and model methods (including computer-based methods) to control system processes and monitor system outputs	√

3.1	Identify and describe the function of the major components of a computer system.
3.2	Use the computer as a tool for generating and drawing ideas.	√
3.3	Control computerized devices and systems through programming.	√
3.4	Model and simulate the design of a complex environment by giving direct commands.

3.1	Assemble a computer system including keyboard, central processing unit and disc drives, mouse, modem, printer, and monitor
3.2	Use a computer system to connect to and access needed information from various Internet sites	√
3.3	Use computer hardware and software to draw and dimension prototypical designs	√
3.4	Use a computer as a modeling tool	√
3.5	Use a computer system to monitor and control external events and/or systems	√

3.1	Understand basic computer architecture and describe the function of computer subsystems and peripheral devices
3.2	Select a computer system that meets personal needs
3.3	Attach a modem to a computer system and telephone line, set up and use communications software, connect to various online networks, including the Internet, and access needed information using email, telnet, gopher, ftp, and web searches
3.4	Use computer-aided drawing and design (CADD) software to model realistic solutions to design problems	√
3.5	Develop an understanding of computer programming and attain some facility in writing computer programs	√

6.1	Describe how outputs of a technological system can be desired, undesired, expected, or unexpected
6.2	Describe through examples how modern technology reduces manufacturing and construction costs and produces more uniform products

6.1	Explain that although technological effects are complex and difficult to predict accurately, humans can control the development and implementation of technology.
6.2	Explain how computers and automation have changed the nature of work
6.3	Explain how national security is dependent upon both military and nonmilitary applications of technology

7.1	Participate in small group projects and in structured group tasks requiring planning, financing, production, quality control, and follow-up.	√
7.2	Speculate on and model possible technological solutions that can improve the safety and quality of the school or community environment.

Intermediate
7.1	Manage time and financial resources in a technological project
7.2	Provide examples of products that are well (and poorly) designed and made, describe their positive and negative attributes, and suggest measures that can be implemented to monitor quality during production
7.3	Assume leadership responsibilities within a structured group activity	√

Commencement
7.1	Develop and use computer-based scheduling and project tracking tools, such as flow charts and graphs
7.2	Explain how statistical process control helps to assure high quality output
7.3	Discuss the role technology has played in the operation of successful U.S. businesses and under what circumstance they are competitive with other countries
7.4	Explain how technological inventions and innovations stimulate economic competitiveness and how, in order for an innovation to lead to commercial success, it must be translated into products and services with marketplace demand
7.5	Describe new management techniques (e.g., computer-aided engineering, computer-integrated manufacturing, total quality management, just-in-time manufacturing), incorporate some of these in a technological endeavor, and explain how they have reduced the length of design-to-manufacture cycles, resulted in more flexible factories, and improved quality and customer satisfaction
7.6	Help to manage a group engaged in planning, designing, implementation, and evaluation of a project to gain understanding of the management dynamics	√

Intermediate
2.1	Apply academic knowledge and skills using an interdisciplinary approach to demonstrate the relevance of how these skills are applied in work-related situations in local, state, national, and international communities
2.2	Solve problems that call for applying academic knowledge and skills	√
2.3	Use academic knowledge and skills in an occupational context, and demonstrate the application of these skills by using a variety of communication techniques (e.g., sign language, pictures, videos, reports, and technology).

Commencement
2.1	Demonstrate the integration and application of academic and occupational skills in their school learning, work, and personal lives.	√
2.2	Use academic knowledge and skills in an occupational context, and demonstrate the application of these skills by using a variety of communication techniques (e.g., sign language, pictures, videos, reports, and technology)	√
2.3	Research, interpret, analyze, and evaluate information and experiences as related to academic knowledge and technical skills when completing a career plan.

1.1	Observe and describe interactions among components of simple systems.	√
1.2	Identify common things that can be considered to be systems (e.g., a plant population, a subway system, human beings).

1.1	Describe the differences between dynamic systems and organizational systems.
1.2	describe the differences and similarities between engineering systems, natural systems, and social systems.
1.3	Describe the differences between open- and closed-loop systems.
1.4	Describe how the output from one part of a system (which can include material, energy, or information) can become the input to other parts.

1.1	Explain how positive feedback and negative feedback have opposite effects on system outputs.
1.2	Use an input-process-output-feedback diagram to model and compare the behavior of natural and engineered systems.
1.3	Define boundary conditions when doing systems analysis to determine what influences a system and how it behaves.

6.1	Choose the best alternative of a set of solutions under given constraints.	√
6.2	Explain the criteria used in selecting a solution orally and in writing	√

6.1	Determine the criteria and constraints and make trade-offs to determine the best decision.	√
6.2	Use graphs of information for a decision-making problem to determine the optimum solution.

2.1	Collect,analyze,interpret,and present data,using appropriate tools	√
2.2	When students participate in an extended,culminating mathematics,science,and technology project, then students should:
	Work effectively—Contributing to the work of a brainstorming group, laboratory partnership, cooperative learning group, or project team; planning procedures; identify and managing responsibilities of team members; and staying on task, whether working alone or as part of a group.	√
	Gather and process information —Accessing information from printed media, electronic data bases, and community resources and using the information to develop a definition of the problem and to research possible solutions.	√
	Generate and analyze ideas — Developing ideas for proposed solutions, investigating ideas, collecting data, and showing relationships and patterns in the data.	√
	Observe common themes—Observing examples of common unifying themes, applying them to the problem, and using them to better understand the dimensions of the problem.	√
	Realize ideas—Constructing components or models, arriving at a solution, and evaluating the result.	√
	Present results—Using a variety of media to present the solution and to communicate the results.	√

2.1	Identify academic knowledge and skills that are required in specific occupations
2.2	Demonstrate the difference between the knowledge of a skill and the ability to use the skill
2.3	Solve problems that call for applying academic knowledge and skills.	√

3b.1	Basic Business Understanding—demonstrate an understanding of business, marketing, and multinational economic concepts, perform business-related mathematical computations, and analyze/interpret business-related numerical information.
3b.2	Business-Related Technology—select, apply, and troubleshoot hardware and software used in the processing of business transactions.
3b.3	Information Management/Communication—prepare, maintain, interpret/analyze, and transmit/ distribute information in a variety of formats while demonstrating the oral, nonverbal, and written communication skills essential for working in today's international service-/information-/technological-based economy.
3b.4	Business Systems—demonstrate an understanding of the interrelatedness of business, social, and economic systems/subsystems.

3b.1	Foundation Development—Develop practical understanding of engineering technology through reading, writing, sample problem solving, and employment experiences.
3b.2	Technology—Demonstrate how all types of engineering/technical organizations, equipment (hardware/software), and well-trained human resources assist and expedite the production/distribution of goods and services
3b.3	Engineering/Industrial Processes—Demonstrate knowledge of planning, product development and utilization, and evaluation that meets the needs of industry.

Who am I as a citizen?	Students develop self-management skills for researching a topic. Students develop critical thinking skills. Students develop effective interpersonal skills.
How are my school experiences connected to my future success?	Students will acquire skills in decision making, communication, and teamwork. Students will learn various management skills. Students will participate in a simulated work environment.
How are my social skills related to my future success?	Students will predict future situations. Students will work as a team to complete a project. Students will develop problem-solving strategies. Students will interact effectively with team partners.
How do I develop the skills and abilities that I need to be successful in a career? How do I find out what I want to know?	Students will research topics using the internet
Why do the choices I make now matter to my future?	Students will gain an awareness of the importance of personal responsibility and good work habits. Students will gain an awareness of the impact of their actions and choices.
How do I affect the systems within which I live and work?

Business/Information Systems (3b)
Basic Business Understanding
Business-Related Technology
Information Management/ Communication
Business Systems
Resource Management
Interpersonal Dynamics
Career Development (1)
Complete development of career plan
Apply decision-making skills in selection of a career option
Analyze skills and abilities in a career option
Integrated Learning (2)
Demonstrate integration and application
Use academic knowledge and skills
Research, interpret, analyze, and evaluate information
Universal Foundation Skills (3a)
Basic Skills
Thinking Skills
Personal Qualities
Interpersonal Skills
Technology
Managing Information
Managing Resources
Systems