Chebyshev Linkage Wheel

This apparatus is a square wheel made by Chebyshev Linkage (approximate straight-line mechanism). In figure, the solid-line bars are components of this apparatus. The dashed-lines are explanatory lines. Aqua-colored big bullet E is a axis of this wheel.
This apparatus has beautiful property. ① ABCD (= #1) is a basic component. AB = Lower base (length 4), AD or BC = Cross-oblique sides (length 5), CD = Upper base (length 2). (We suppose that AD and BC are always crossed. To avoid the uncrossed situation , we must attach the angle limitation stopper. i.e. Keep BP ≧ 0.) This apparatus has 4 components (#1,#2,#3,#4). #1 and #3 are point-symmetry each other. #2 and #4 are, too. ② #1 and #2 (LM, ON) position relation has next rule. a) Lower base ⊥ others Upper base. From a), ON (#2 Upper base) is always vertical. ∵ AB is fixed and horizontal. From a), LM (#2 Lower base) is rotate 0 to 180゜. ∵ CD rotates 180゜. [ In same component, Lower and Upper bases move alternately each other. i.e. When one is rotating, the other is not-rotating. ] Rule a) is established by various ways, but, the most simple way is used in above figure. i.e. R = the middle of BC (length 2.5 [= 5/2] ), Suppose S , RS = length 2.5, ES= length 2 [= 4/2]. From theoretical analysis, ES // AB. --- so, To make ON ⊥ AB is equivalent to to make ON ⊥ ES. (We can reduce the number of bar by this way.) At the end, Force to fix the angle between ES and ON toward 90゜. --- welding/ glue. In real implementation, prepare short macaroni tube== (plural pasrts), And insert axis bar --- into it. as --==---==---==---==---. ← 4 macaroni. Axis design: ---- Implementation is easy. ( #1 is independent, others are all dependent.) #1/#3 upper-bar ①  ---- == macaroni_1 (bonds with naked Axis-bar ---) #1/#3 lower-bar ES ②  ---- == macaroni_2 #2/#4 upper-bar ③ (⊥ with ②)  ---- == macaroni_3 (bonds with macaroni_2) #2/#4 lower-bar EU ④ (⊥ with ①)   ---- == macaroni_4 (bonds with naked Axis-bar ---) Chassis (car_body) ---- == macaroni_5 [ i.e. The smallest configure is ------==-- style. With only one macaroni. ] I think this inventions is a home-run hit in the human being history. (This figure was made by GeoGebra Graphics View's copy-function. GeoGebra is powerful educational tool.) ★ Feeling. Please imagine, there exists vertical real gravity ↓ [vector ON], so, ground AB is horizontal. upper base CD vector is 2nd. virtual gravity →, so, #2 Lower base MO' ⊥ CD. virtual gravity rotates, in last, it directs to real gravity direction. and, it will be baton touched/ passed. ■ Easy making (??)  ---- This is a kind of instant wheel. 1. Prepare 2 sets of Chebyshev point-symmetry linkages. 2. Prepare one axis bar. 3. Insert the axis bar into 2 each symmetry points. Here, axis bar is like a BBQ skewer. 4. attach the rule a) cordinator 3 bars (2.5, 2.5, 2 length bars) between to 2 point-symmetry linkages.  5. Set 90° adapter (corresponds to α = 90° mechanism ).  ■ About practical utility : ・ wheel for on a sand-beach, carpet (----- small loss caterpillar. ) ・ caster of travel bag ・ instant wheel ( They are only bars, usually.) ・ heavy weight lifting wheel/ roller (constant height) cf. Monster 4x4 Mud Trucks at Good Time 4x4's June 12Th 2011 (YouTube) ---- interesting vtr. ★ Variant :Q: Please consider Hexagon (= 6 edges) wheel. 3 sets of the point symmetry Chebyshev frame. A: It's easy. ---- rule a') Lower base ∠60° & ∠120° other 2 Upper bases. (2 grounded foot width is narrow than 4 edgee case.) cf. Chebyshev Linkage Wheel2 ■ Pedal driven implementation FIG. Below pink square is available to make a pedal crank mechanism.