©1998, 2011, 2015, 2017 Raffi J. Kasparian
Nonregular Star Polyhedra
Articles of Interest:
True-To-Life 3D Object Rotations Using A Mouse
What is a Kasparian Construction?
Archimedean opens with "Create..." selected. Select "Create..." to create any solid that you can define.
Select "Transform..." to transform created solids by various predefined methods.
Select "Analyze..." to interactively view various properties of the created solid.
Select "Options..." to vary the method by which Archimedean displays solids.
Archimedean provides a set of video controls to allow you to analytically view the assembly of a solid.
Transform...Platonic and Archimedean solids are distinguished by the pattern of polygonal sides around any corner. For instance, a cube has three square faces at every corner (4 4 4) while a truncated cube has two octagons and a triangle at every corner (8 8 3). Unless Predefined Solids is selected, you must define a corner of the solid you wish to view before Archimedean can create it. Use the Polygon Palette to define a corner. When you have defined the corner, click on the Create button for Archimedean to create the solid.Predefined Solids (Default: OFF)Select for a list by name of all Platonic, Archimedean, Poinsot and Rational solids as well as their duals (except for Poinsot Duals). Click on the name of a solid to create that solid. While solids other than these may be created, they must be entered with the polygon palette. For instance, a pentagonal prism would be entered by choosing a square twice and a pentagon once (4 4 5).Polygon PaletteSelect a polygon from the Polygon Palette to add another side to a corner when defining a corner in Create Mode or when adding a side to a partially built solid in Manual Mode.New...
CircuitsSelect the number of times polygons should circumnavigate a vertex. Poinsot solids require 2 circuits. Other solids require 1. Up to four are allowed.Polygon ButtonsClick a polygon button to select the depicted polygon.Other...Choose to select a polygon unavailable through the polygon buttons. Enter the number of sides of the desired polygon and click "OK".Select to start the creation process over.CreateWhen you have chosen the sides you would like at the corner of the solid, click this to tell Archimedean to create the solid.
Dual...Analyze...Every Platonic, Archimedean, Poinsot and Rational solid has a dual. The dual is the solid created by replacing each corner of the original solid with a side whose corners are the centers of the sides that surrounded the original corner. Another way to look at it is that the center of every side of the original solid is replaced by a corner of the dual. The dual of a Platonic solid is another Platonic solid. The dual of the dual of a Platonic solid is the original Platonic solid. The dual of an Archimedean, Poinsot or Rational solid is neither Archimedean or Platonic and does not have a dual.Antidual...
Show DualReveals the dual of the current solid.Show OriginalReveals the original solid.Available only when the dual of an Archimedean solid is the current solid and New Transformation has been selected. The antidual reverses the dual process to return the current solid to the original Archimedean solid.Truncate Corners...
Show AntidualReveals the antidual of the current Archimedean dual.Show OriginalReveals the original solid.Uniformly slices off every corner of the solid. Use the slider to vary the amount of truncation. As the degree of truncation increases, Platonic solids pass through three different Archimedean solid stages before ariving at the final stage which is simply the inscribed dual. Click on a stage name to bring the solid to that degree of truncation.Truncate Edges...Uniformly slices off every edge of the solid. Use the slider to vary the amount of truncation. Archimedean solids pass through a varying number of stages before arriving at their final stages. Click on a stage name to bring the solid to that degree of truncation.Resize...Resizes the solid.New TransformationSelect to treat a transformed solid as the starting point for further transformations. Note that not all solids can be transformed further.
ClassBasic category of solid: Platonic, Archimedean, Dual, Poinsot, Rational or Other.NameSpecific name of solid.CornersClick to highlight the corners. Click again to unhighlight the corners. Click any subcategory to highlight or unhighlight that subcategory of corners.SidesClick to reveal all sides. Click any subcategory to reveal only that subcategory of sides.EdgesThe number of edges.Options...Fill (Default: ON)Make every side opaque.Draw Edges (Default: ON)Highlight the edges.Color Code (Default: ON)Every side is given a different color based on the number of edges in the side. Available only when Fill is enabled.Draw (Default: OFF)Make every side transparent.1st ColorThe color an original solid will be drawn in. Use the slider to dynamically change this color.2nd ColorThe color any new sides arising from a transformation will be drawn in. Useful when studying the truncation process. Use the slider to dynamically change this color. Available only when Fill is enabled.Gray Scale (Default: OFF)Render solids in gray scale.Show Coordinates (Default: OFF)
Select to display the 3D coordinates of the cursor location. the x-axis is horizontal and increases to the right, the y-axis is the depth axis and increases as it moves away from the viewer, and the z-axis is vertical and increases as it moves up. The 3D origin is on the screen surface at the center of Archimedean's viewing area.Using the Video Controls
|<<Rewind the assembly process to the beginning.<<Rewind the assembly process step by step to the beginning.<Rewind the assembly process by one step.||Pause/Resume the assembly at the current point in the process.>Forward the assembly process by one step.>>Forward the assembly process step by step to the end.>>|Forward the assembly process to the end.Using the MouseClick any where on the solid to select a surface point of the solid and then drag the mouse to rotate the solid by this surface point. Shift-click and drag to constrain rotation to the x- or z-axis. Click outside the surface of the solid to rotate around the y-axis. Rotation is around the origin. Solids are centered on the origin. The cursor changes depending on whether the selected point is on the front or back side of the solid. When you drag the cursor outside the bounds of the solid, Archimedean automatically considers the cursor to be on the other side. This allows you to rotate the solid full circle by dragging the cursor back and forth in and out of the solid's bounds.DefinitionsPlatonic SolidDiscovered by Plato. A solid composed of regular polygonal sides of one type having a uniform pattern of sides around each corner.Archimedean SolidDiscovered by Archimedes. A solid composed of regular polygonal sides of two or more types having a uniform pattern of sides around each corner.DualA transformation of a regular uniform polyhedron whereby (in effect) vertices are replaced by polygons and polygons by vertices.Poinsot SolidOne of two solids discovered by Louis Poinsot where each vertex is formed by polygons circumnavigating that vertex twice.Rational SolidSolids discovered by Raffi Kasparian similar to Archimedean solids except that polygons circumnavigate the solid more than once.
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