Geometer s Sketchpad can do the same thing but still has the power to manipulate the unit piece after you have completed your tessellated plane.

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Giles Curtis
5 years ago
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1 (Gemeter's Sketchpad Tessellatins) Have yu ever made a tessellatin ut f fragmented parallelgram? Gemeter s Sketchpad can d the same thing but still has the pwer t manipulate the unit piece after yu have cmpleted yur tessellated plane. Start by cnnecting several little line segments tgether t frm the bttm and left side f the parallelgram as shwn at the right. (Please make certain that each adjacent segments share the same endpints) Label the 3 pints that will becme the vertices f the parallelgram with the finger tl. Select Pint A and I in that rder. Select Mark Vectr A >I under the Transfrm menu. Watch yur sketch as yu select this frm the menu and yu shuld see a little animatin f a dtted 'laser beam' ging frm A t I. With the selectin tl drag a bx arund all f the segments frm A t E as shwn at the right After the fragmented segment is highlighted, select Translate frm the Transfrm menu. Make sure the radi buttn labeled By Marked Vectr is checked. Then, click Translate. This will create the tp segment f yur fractured parallelgram. An easy way t select many bjects at nce is t "drag a bx" arund the items yu wish t be selected as shwn belw. Next, we will need t mark the vectr AE. Using the selectin tl, select Pint A and E in that rder. Select Mark Vectr A->E under the Transfrm menu. Again, watch yur sketch as yu select this frm the menu and yu shuld see a little animatin f a dtted 'laser beam' ging frm A t E. M. Winking p.70 wisene@mindspring.cm

$Make sure the radi buttn labeled By Marked Vectr is checked. Then, click Translate. This will create the right hand side segment f yur fractured parallelgram.$ $Next hld dwn shift (this will help incase yu miss clicking n a pint yu wn t deselect everything) and begin clicking n each pint arund the perimeter f the fractured parallelgram in cnsecutive rder ($ We must click n each pint as we wuld if we were shwing smene hw t cnnect the dts t create the parallelgram). After all f the pints are selected, under the Cnstruct menu select Plygn interir.

We must click n each pint as we wuld if we were shwing smene hw t cnnect the dts t create the parallelgram). After all f the pints are selected, under the Cnstruct menu select Plygn interir.

2 With the selectin tl we will need t carefully need t drag a bx arund all f the segments between A and I Once Again, after the fragmented segment is highlighted, select Translate frm the Transfrm menu. Make sure the radi buttn labeled By Marked Vectr is checked. Then, click Translate. This will create the right hand side segment f yur fractured parallelgram. Using the selectin tl, deselect everything by clicking in any blank space f the sketch. Next hld dwn shift (this will help incase yu miss clicking n a pint yu wn t deselect everything) and begin clicking n each pint arund the perimeter f the fractured parallelgram in cnsecutive rder ( We must click n each pint as we wuld if we were shwing smene hw t cnnect the dts t create the parallelgram). After all f the pints are selected, under the Cnstruct menu select Plygn interir. This will fill in the interir f the plygn. While the plygn interir is still highlighted under the Display menu we can and s n all the way arund the parallelgram change the plygn s interir clr. The last vectr we marked was AE. We can re-use this marking t translate ur plygn interir. Make sure the plygn interir is still highlighted. select Translate frm the Transfrm menu and click Translate This will create a new plygn interir directly t the right f the unit piece f ur tessellatin. While the newly created plygn is still highlighted, change its clr under the Display menu and then, select Translate frm the Transfrm menu. Repeat this prcess until there are abut 5 8 parallelgrams acrss the bttm. (The tessellatin may lk best with alternating clrs as shwn at the right) We can chse the clr f the interir f the plygn under the Display menu. T better establish alternating clrs we will need t mark a new vectr EI. (It may first be necessary t deselect everything by clicking in a blank area with the selectin tl) Using the selectin tl, select Pint E and I in that rder. Select Mark Vectr E->I under the Transfrm menu. Again, watch yur sketch as yu select this frm the menu and yu shuld see a little animatin f a dtted 'laser beam' ging frm E t I. Using the selectin tl, select all f the plygn interirs by clicking n each f them. After all f the interirs are selected, under the Transfrm menu select Translate and click Translate While the new rw f plygn interirs are still highlighted, again select Translate frm the Transfrm menu. Repeat this prcess until there are abut 5 8 rws. Hw d yu think we might cntinue t tessellate this space up and t the right? Which vectr shuld we mark? Which plygn interirs shuld we highlight and then translate? DO NOT frget the mst imprtant aspect f Sketchpad, "Everything is dynamic." Play arund with the pints n the unit piece by clicking and dragging the pints. Experiment with setting tracks fr riginal pints and animating. M. Winking p.71 wisene@mindspring.cm

3 Have yu ever made a tessellatin ut f fragmented equilateral triangle? Gemeter s Sketchpad can d the same thing but still has the pwer t manipulate the unit piece after yu have cmpleted yur tessellated plane. Start by creating a several cnjined segments that wuld represent the fragmented left side f the equilateral triangle. Duble click n the tp pint (pint B) f the fractured segment. This will mark the pint as the center f rtatin. Highlight the entire fragmented side f the triangle by dragging a bx arund the entire side. Select Rtate under the Transfrm menu. Change the angle t 60º and press the Rtate buttn. Using the segment tl create a segment between the tw bttm pints which shuld represent the tw bttm vertices f the equilateral triangle. While the segment is still highlighted, select Midpint frm the Cnstruct menu. Click in a blank area t deselect everything. Highlight the bttm segment and select Hide Segment under the Display menu. The midpint shuld remain. M. Winking p.72 wisene@mindspring.cm

Make a set f cnjined segments between pints A and D. Duble click n the midpint (pint D) until it flashes denting that it has been marked as a center.

$Next hld dwn shift (this will help incase yu miss clicking n a pint yu wn t deselect everything) and begin clicking n each pint arund the perimeter f the fractured triangle in cnsecutive rder ( We$ must click n each pint as we wuld if we were shwing smene hw t cnnect the dts t create the triagnle). After all f the pints are selected, under the Cnstruct menu select Plygn interir.

must click n each pint as we wuld if we were shwing smene hw t cnnect the dts t create the triagnle). After all f the pints are selected, under the Cnstruct menu select Plygn interir.

4 Make a set f cnjined segments between pints A and D. Duble click n the midpint (pint D) until it flashes denting that it has been marked as a center. Highlight all f the cnjined segments between pint A and D by dragging a bx arund them. Then, select Rtate under the Transfrm menu. Change the angle t 180º and press the rtate buttn. Using the selectin tl, deselect everything by clicking in any blank space f the sketch. Next hld dwn shift (this will help incase yu miss clicking n a pint yu wn t deselect everything) and begin clicking n each pint arund the perimeter f the fractured triangle in cnsecutive rder ( We must click n each pint as we wuld if we were shwing smene hw t cnnect the dts t create the triagnle). After all f the pints are selected, under the Cnstruct menu select Plygn interir. This will fill in the interir f the plygn and s n all the way arund the triangle Click in a blank space t deselect everything. Then, duble click n the tp vertex (pint B). After marking pint B as the center f a rtatin, highlight the plygn interir. Select Rtate under the Transfrm menu. Change the angle t 60º and press the Rtate buttn. While the new image is highlighted, select Rtate again frm the Transfrm menu. Cntinue t use an angle f 60º and press the Rtate buttn again. Repeat this prcess until the entire tessellated hexagn is cmpleted (3 mre times). When cmplete, yu may wish t alternate the clrs. Experiment with the clrs under the Display menu. Next duble click n a vertex f the hexagn (such as pint A). Highlight all 6 plygn interirs and select Rtate frm the Transfrm menu. Change the angle t 120º and press the Rtate buttn. While the new image is still highlighted, rtate the new image anther 120º in a similar manner. Remember t play with all f the riginal pints. Try experimenting with animating the unit piece. M. Winking p.73 wisene@mindspring.cm

Nn Traditinal Tessellatins (Escher-esque) Start by creating a circle similar t the ne shwn at the right. Then, create tw radii f the circle similar t the diagram at the right.

Using the selectins tl, highlight radius AB, radius AC and the Circle. Select Hide Path Objects frm the Display menu. Only the pints shuld remain.

Select Mark Angle under the Transfrm menu. Yu shuld see an animatin n the sketch denting the angle has been marked.

Make certain the radi buttn is checked next t Marked Angle BAC abut Center A. Click n the buttn Rtate. Using the segment tl, create a set f cnjined segments between B and C.

5 Nn Traditinal Tessellatins (Escher-esque) Start by creating a circle similar t the ne shwn at the right. Then, create tw radii f the circle similar t the diagram at the right. Using the pint tl, place a pint n radius AB smewhere twards the edge f the circle. Label this pint D. Click in a blank space t deselect everything. Using the selectins tl, highlight radius AB, radius AC and the Circle. Select Hide Path Objects frm the Display menu. Only the pints shuld remain. Using the segment tl, create a set f cnjined segments between B and D. Click in a blank area t deselect everything. Using the selectin tl, highlight Pint B, A, and C in that rder. Select Mark Angle under the Transfrm menu. Yu shuld see an animatin n the sketch denting the angle has been marked. Drag a bx arund the cnjined segment and all f the pints between them except fr pint B (shwn at the right). Select Rtate under the Transfrm menu. Make certain the radi buttn is checked next t Marked Angle BAC abut Center A. Click n the buttn Rtate. Using the segment tl, create a set f cnjined segments between B and C. Click in a blank area t deselect everything. Using the selectin tl, highlight Pint A, B, and D in that rder. Select Mark Rati under the Transfrm menu. Yu shuld see an animatin n the sketch denting the angle has been marked. Drag a bx arund the cnjined segments between pints B and C (shwn at the right). Select Dilate under the Transfrm menu. Make certain the radi buttn is checked next t Marked Rati (AD/AB) abut Center A. Click n the buttn Dilate. Using the selectin tl, deselect everything by clicking in any blank space f the sketch. Next hld dwn shift (this will help incase yu miss clicking n a pint yu wn t deselect everything) and begin clicking n each pint arund the perimeter f the fractured quadrilateral in cnsecutive rder ( We must click n each pint as we wuld if we were shwing smene hw t cnnect the dts t create the quadrilateral) and s n all the way arund the quadrilateral After all f the pints are selected, under the Cnstruct menu select Plygn interir. This will fill in the interir f the plygn M. Winking p.74 wisene@mindspring.cm

While the plygn interir is still highlighted select Dilate frm the Transfrm menu. Make certain the radi buttn is still checked next t Marked Rati (AD/AB) abut Center A. Click n the buttn Dilate.

After enugh images have been generated, try selecting every ther plygn and changing the clr under the Display menu. Using the selectin tl, highlight all f the plygn interirs and Pint C.

The clrs will f the new images will need t be changed s that they alternate prperly. E After crrecting the clr sequence, select Pints B, A, and E in that rder.

6 While the plygn interir is still highlighted select Dilate frm the Transfrm menu. Make certain the radi buttn is still checked next t Marked Rati (AD/AB) abut Center A. Click n the buttn Dilate. While the new image f the plygn is highlighted select Dilate again and cntinue dilating a few mre images until the images get relatively clse t pint A. After enugh images have been generated, try selecting every ther plygn and changing the clr under the Display menu. Using the selectin tl, highlight all f the plygn interirs and Pint C. Select Rtate under the Transfrm menu. Make certain the radi buttn is checked next t Marked Angle BAC abut Center A. Click n the buttn Rtate. The clrs will f the new images will need t be changed s that they alternate prperly. E After crrecting the clr sequence, select Pints B, A, and E in that rder. Select Mark Angle under the Transfrm menu. Highlight all f the plygn interirs and select Rtate under the Transfrm menu. Make certain the radi buttn is checked next t Marked Angle BAE abut Center A. Click n the buttn Rtate. Cntinue rtating each new image t cmplete the circle. Dn t frget t alter the pre-image pints. Try experimenting with cpying and pasting sme f M.C. Escher s wrk int Gemeter s Sketchpad. Can yu find investigate hw he made sme f his mre difficult hyperblic tessellatins? M. Winking p.75 wisene@mindspring.cm

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