This apparatus is a square wheel made by [b]Chebyshev Linkage[/b] (approximate straight-line mechanism).[br][br]In figure, the solid-line bars are components of this apparatus. The dashed-lines are explanatory lines.[br][br][color=#00ffff]Aqua[/color]-colored big bullet [b][color=#00ffff][size=150]●[/size][/color] [color=#00ffff]E[/color][/b] is a axis of this wheel.

This apparatus has beautiful property.[br]① ABCD (= #1) is a basic component. [br]AB = Lower base (length 4), AD or BC = Cross-oblique sides (length 5), CD = Upper base (length 2).[br](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.)[br][br]This apparatus has 4 components (#1,#2,#3,#4). #1 and #3 are [b]point-symmetry[/b] each other. #2 and #4 are, too.[br]② #1 and #2 (LM, ON) position relation has next [b]rule[/b].[br][b]a) Lower base ⊥ others Upper base.[/b][br][br]From a), ON (#2 Upper base) is always vertical. ∵ AB is fixed and horizontal. [br]From a), LM (#2 Lower base) is rotate 0 to 180゜. ∵ CD rotates 180゜. [br][ In same component, Lower and Upper bases move alternately each other. i.e. When one is rotating, the other is not-rotating. ][br][br][b]Rule a)[/b] is established by various ways, but, the most simple way is used in above figure.[br]i.e. R = the middle of BC (length 2.5 [= 5/2] ), Suppose S , RS = length 2.5, ES= length 2 [= 4/2].[br]From theoretical analysis, ES // AB. --- so, To make ON ⊥ AB is equivalent to to make ON ⊥ ES.[br](We can reduce the number of bar by this way.)[br]At the end, Force to fix the angle between ES and ON toward 90゜. --- welding/ glue.[br][b]In real implementation, prepare short [/b][color=#0000ff]macaroni tube[/color][color=#0000ff]==[/color][b] (plural pasrts),[br]And insert axis bar [/b][color=#ff00ff]---[/color][b] into it. [br]as [/b][color=#ff00ff]--[/color][color=#0000ff]==[/color][color=#ff00ff]---[/color][color=#0000ff]==[/color][color=#ff00ff]---[/color][color=#0000ff]==[/color][color=#ff00ff]---[/color][color=#0000ff]==[/color][color=#ff00ff]---[/color][b]. ← 4 macaroni.[br][/b][color=#ff0000][br][b]Axis design: ---- Implementation is easy.[/b]　[/color]( #1 is independent, others are all dependent.)[br][color=#38761d]#1/#3 upper-bar ①　　---- [/color][color=#0000ff]==[/color][color=#38761d] macaroni_1 (bonds with naked Axis-bar [b][color=#ff00ff]---[/color][/b])[br][color=#38761d]#1/#3[/color] lower-bar ES ②　　---- [/color][color=#0000ff]==[/color][color=#38761d] macaroni_2[br]#2/#4 upper-bar ③ (⊥ with ②)　　---- [/color][color=#0000ff]==[/color][color=#38761d] macaroni_3 (bonds with macaroni_2)[br]#2/#4 lower-bar EU ④ (⊥ with ①)　　　---- [/color][color=#0000ff]==[/color][color=#38761d] macaroni_4 (bonds with naked Axis-bar [b][color=#ff00ff]---[/color][/b])[br]Chassis (car_body) [/color][color=#38761d]---- [/color][color=#0000ff]==[/color][color=#38761d] macaroni_5[br][/color][ i.e. The smallest configure is [b][color=#ff00ff]------[/color][color=#0000ff]==[/color][color=#ff00ff]--[/color][/b] style. With only one macaroni. ][br][br]I think this inventions is a home-run hit in the human being history.[br](This figure was made by GeoGebra Graphics View's copy-function. GeoGebra is powerful educational tool.)[br][br][b]★ Feeling.[/b][br]Please imagine, there exists vertical real gravity ↓ [vector ON], so, ground AB is horizontal. [br]upper base CD vector is 2nd. virtual gravity →, so, #2 Lower base MO' ⊥ CD.[br]virtual gravity rotates, in last, it directs to real gravity direction. and, it will be baton touched/ passed. [br][br][b]■ Easy making (??)  ---- This is a kind of instant wheel.[/b][br]1. Prepare 2 sets of Chebyshev point-symmetry linkages. [br]2. Prepare one axis bar.[br]3. Insert the axis bar into 2 each symmetry points. Here, axis bar is like a BBQ skewer.[br]4. attach the [b]rule a)[/b] cordinator 3 bars (2.5, 2.5, 2 length bars) between to 2 point-symmetry linkages. [br]5. Set 90° adapter (corresponds to α = 90° mechanism ). [br][br][b]■ About practical utility :[/b][br]・ wheel for on a sand-beach, carpet (----- small loss caterpillar. )[br]・ caster of travel bag[br]・ instant wheel ( They are only bars, usually.)[br]・ heavy weight lifting wheel/ roller (constant height)[br]cf. [url=https://www.youtube.com/watch?v=G837cDBvQe8]Monster 4x4 Mud Trucks at Good Time 4x4's June 12Th 2011[/url] (YouTube)　---- interesting vtr.[br][br][b]★ Variant :[/b]Q: Please consider Hexagon (= 6 edges) wheel. 3 sets of the point symmetry Chebyshev frame.[br][br]A: It's easy. ---- rule a') Lower base ∠60° & ∠120° other 2 Upper bases.[br](2 grounded foot width is narrow than 4 edgee case.)[br][br]cf. [url=https://www.geogebra.org/m/cRajdZcP?doneurl=%2Fasifsound]Chebyshev Linkage Wheel2[/url] [br][br][size=150][size=200][b]■ Pedal driven implementation FIG.[br][/b][/size][/size]Below [color=#ff00ff][b]pink square[/b][/color] is available to make a pedal crank mechanism.