The courses show how both the Old and New Testaments point toward Jesus. The entire series will be focused primarily upon Jesus Christ, drawing from the entire Bible for each course. The study for older students is probably best used for independent study, although it can also be used with a group class.Īll of the books are printed in full color and have plastic spiral bindings so the books will lie flat. With the younger-level study you can teach all of your children in the elementary grades together. This new series consists of two courses thus far-one that can be used with students in grades one through six and the other for grades seven through twelve. (The publisher’s statement of faith is more generic and limited than the theological positions taught in these courses.) I point this out because both courses that I reviewed have strong theological content that is often deeper than what you find in other Bible study courses for these levels, and many of my readers will want to know what denominational viewpoints they are likely to support. The series is written from a viewpoint that tends to reflect Reformed Protestant doctrine, although it is not intended to advance any particular denomination. The Illuminate Bible series offers homeschooling families a resource for in-depth Bible study. Online Schools with Complete Programs and Courses: Secular.Online Schools with Complete Programs and Courses: Religious.Standardized and Special Needs Testing Products.Reading and Phonics Parent Resources and Other Helps.Publishers Offering Courses for Many Languages.Ungraded, Multi-level Resources - Composition.Ungraded, Multi-level Resources - Grammar.Ungraded, Multi-level Resources - Comprehensive.Bible & Religion Parent & Family Resources."Illumination Problem."įrom MathWorld-A Wolfram Web Resource. On Wolfram|Alpha Illumination Problem Cite this as: "Polygonal Rooms Not Illuminableįrom Every Point." Amer. "Is Every Polygonal Region Illuminable from Some Point?" Math. In the Season 4 opening episode " Trust Metric" (2007) of the television crime drama NUMB3RS, math genius CharlieĮppes uses the illumination problem and Tokarsky's 26-sided room as the motivation Of reflection-which states that the angle of incidence equals the angle of reflection-applies Reflection of a billiard ball from a cushion of a billiard table because the law The reflection of light from the surface of a mirror is exactly analogous to the Illumination problems are intimately related to billiards. Light source is placed at a given position. In actuality a finite number of dark points remain unilluminated when a point It should be noted that the Tokarsky room is a borderline case, because Tokarsky (1995 left figure) constructed a 26-sided polygonal counterexample to (1) in the plane, which was subsequently improved to a 24-sided room (Castro 1997 rightįigure). Tokarsky (1995) showed that unilluminable polygonal rooms exist in the plane and three dimensions, but question (2) remains open in the case of polygonal rooms. Space within the blue border) can never be fully illuminated. Source is indicated by the black cross-hairs. In white, unilluminated regions are indicated in gray, and the position of the light In this figure, lit regions are indicated Three possible configurations of illumination. The ellipses and mushrooms are strategically placedĪs shown, with the red points being the foci of the half-ellipses. (which are in turn built up from straight line segments and smaller half-ellipses) Penrose's room, illustrated above,Ĭonsists of two half-ellipses at the top and bottom and two mushroom-shaped protuberances Regions, regardless of the position of the candle. In 1958, a young Roger Penrose used the properties of the ellipse to describe a room with curved walls that would always have dark (unilluminated) Here, illuminable means that there is a path from every point to every other by repeated reflections. Is every region illuminable from at least one point in the region? Is every region illuminable from every point in the region?Ģ.
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