dezz unit bw - gathering 4 gardner...mitsunobu sonobe, not to mention tom hull’s famous phizz...
TRANSCRIPT
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12. The elevated icosidodecahedrontakes 60 units. Leonardo da Vincidescribed (and named) this solid.
DeZZ UnitCopyright ©2012 by Robert J. Lang
This is actually several units in one: a Deltahedron Zig-Zag unit, which can be used to fold any deltahedron (anypolyhedron whose faces are equilateral triangles). A variation of the unit lets you fold a twisted-hole cube; anothervariation works for any deltahedrally elevated polyhedron; another variation folds a rhomboidal polyhedron that isthe Wolfram Alpha logo. The units draw upon concepts identified and explored by Bob Neale, Lewis Simon, andMitsunobu Sonobe, not to mention Tom Hull’s famous PHiZZ unit, which provides, as well, the rationale for thismodule’s name.
1. Begin with a square,colored side up. Fold inhalf vertically andunfold, making a pinchat the left.
2. Fold the bottom leftcorner to the mark youjust made, creasing aslightly as possible.
3. Fold and unfold alongan angle bisector,making a pinch along theright edge.
4. Fold the top left cornerdown to the creaseintersection.
black dot
hollow dot
arrow tiny
arrow small
arrow large
turn over
rotate cw
rotate ccw
view from here
repeat ll
repeat lr
repeat ul
repeat ur
right angle
cut here
3. Fold and unfold. Repeat behind.
4. Fold and unfold. Repeat behind. 5. Unfold so that the coloredside is visible.
6. The mountain creases here showthe folds that are used for thedeltahedron. Fold 3N/2 units for adeltahedron with N faces.
7. Here is how two units gotogether. The tab slips into thepocket on the back side, and thetwo points marked by dots cometogether.
8. The third unit goes insideone pocket and outside onetab.
9. The dots are the corners of atriangular face. Keep adding unitsto create any deltahedron.
10. Here is a tetrahedron, from 6units.
5. Turn the paper over. 6. Fold the top foldededge down to the rawbottom edge.
7. Fold the bottom foldededge (but not the raw edgebehind it) up to the top andunfold.
8. Fold the top left corner andthe bottom right corner to thecrease you just made.
9. Here’s the building block. 10. Here’s the other side.
Plain Twisted-Hole CubeThis structure is similar to Lewis Simon’s many twist-hole cubes, but uses the assembly technique of RobertNeale’s dodecahedron.
1. Begin with step 9 of the DeZZbuilding block. Fold thequadrilateral in half along thediagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
3. Join three units at a corner bysliding tabs into pockets.
4. The finished cube.
DeltahedraThis unit can be used to make anypolyhedron whose faces are equilateraltriangles.
1. Begin with step 9 of the DeZZbuilding block. Fold the quadrilateralin half along the diagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
11. An octahedron takes 12 units. 12. And an icosahedron takes 30units.
13. With 210 units, one can make adeltahedrified snub dodecahedron.However, the very shallow angles meansthat it doesn’t hold together very well.
Deltahedrally Elevated PolyhedraElevation is the result of erecting a pyramid on each face. If the resulting new faces are equilateraltriangles, then we can fold them from still another version of this unit that makes each face aseamless equilateral triangle.
1. Begin with step 10 of the DeZZbuilding block. Bring two pointstogether.
2. Fold the upper lefttriangle behind and thebottom triangle up.
3. Partially unfoldall of the foldsalong the strip.
4. One unit makesa portion of adouble pyramid.
5. 4 units makes anelevated tetrahedron,which resembles a caltrop.
6. The 12-unit elevatedoctahedron is also astellation of theoctahedron; Kepler calledit the Stella Octangula.
8. The elevated cube takes12 units, and resembles aslightly stubbier version ofthe origami model calledthe Jackstone.
9. The 30-unit elevateddodecahedron is a slightly bumpyball that is close to, but not exactly,a rhombic triacontahedron.
10. The 30-unit elevated icosahedronis considerably bumpier. It is closeto, but not exactly, a stellation of thedodecahedron.
11. The elevated cuboctahedrontakes 24 units.
13. And finally, 200 units will buildyou the elevated smallrhombicosidodecahedron.
14. If you make 3 units with allmountain folds, they can beassembled into the deltahedralequivalent of Takahama’s Jewel.
15 . Which is a deltahedrallyelevated trigonal dihedron, or,more simply, a tetrahedraldipyramid.
18. If a polyhedron is elevatedwith negative height, we call it“depressed.” You can folddepressed polyhedra by changingthe parity of some of the creaseslike this.
19 . This depressed dodecahedronrequires 30 units, like its elevatedkin.
20. Leaving out the middle creasegives a unit that has an interestingapplication...
21. If you elevate the triangles anddepress the pentagons of anicosidodecahedron, you get thisshape, which also happens to bethe logo of Wolfram | Alpha.
23. If you depress the trianglesand elevate the pentagons of theicosidodecahedron, you get anicosahedron with holes.
22. With the mountain fold backin place, we can make other mixedelevated/depressed polyhedra...
24. We can treat the cuboctahedronsimilarly. Elevated triangles, depressedsquares in a cuboctahedron.
25. Elevated squares, depressedtriangles in a cuboctahedron.
16 . Four units gives adeltahedrally elevated squaredihedron, or a square dipyramid,or simply, an octahedron.
17. And 5 such units gives apentagonal dipyramid.
26. And finally, to wrap up, going back to the originalelevated unit, 210 units give a deltahedrally-elevateddeltahedrified snub dodecahedron. (Yes, that’s double-deltahedrification!)
-
12. The elevated icosidodecahedrontakes 60 units. Leonardo da Vincidescribed (and named) this solid.
DeZZ UnitCopyright ©2012 by Robert J. Lang
This is actually several units in one: a Deltahedron Zig-Zag unit, which can be used to fold any deltahedron (anypolyhedron whose faces are equilateral triangles). A variation of the unit lets you fold a twisted-hole cube; anothervariation works for any deltahedrally elevated polyhedron; another variation folds a rhomboidal polyhedron that isthe Wolfram Alpha logo. The units draw upon concepts identified and explored by Bob Neale, Lewis Simon, andMitsunobu Sonobe, not to mention Tom Hull’s famous PHiZZ unit, which provides, as well, the rationale for thismodule’s name.
1. Begin with a square,colored side up. Fold inhalf vertically andunfold, making a pinchat the left.
2. Fold the bottom leftcorner to the mark youjust made, creasing aslightly as possible.
3. Fold and unfold alongan angle bisector,making a pinch along theright edge.
4. Fold the top left cornerdown to the creaseintersection.
black dot
hollow dot
arrow tiny
arrow small
arrow large
turn over
rotate cw
rotate ccw
view from here
repeat ll
repeat lr
repeat ul
repeat ur
right angle
cut here
3. Fold and unfold. Repeat behind.
4. Fold and unfold. Repeat behind. 5. Unfold so that the coloredside is visible.
6. The mountain creases here showthe folds that are used for thedeltahedron. Fold 3N/2 units for adeltahedron with N faces.
7. Here is how two units gotogether. The tab slips into thepocket on the back side, and thetwo points marked by dots cometogether.
8. The third unit goes insideone pocket and outside onetab.
9. The dots are the corners of atriangular face. Keep adding unitsto create any deltahedron.
10. Here is a tetrahedron, from 6units.
5. Turn the paper over. 6. Fold the top foldededge down to the rawbottom edge.
7. Fold the bottom foldededge (but not the raw edgebehind it) up to the top andunfold.
8. Fold the top left corner andthe bottom right corner to thecrease you just made.
9. Here’s the building block. 10. Here’s the other side.
Plain Twisted-Hole CubeThis structure is similar to Lewis Simon’s many twist-hole cubes, but uses the assembly technique of RobertNeale’s dodecahedron.
1. Begin with step 9 of the DeZZbuilding block. Fold thequadrilateral in half along thediagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
3. Join three units at a corner bysliding tabs into pockets.
4. The finished cube.
DeltahedraThis unit can be used to make anypolyhedron whose faces are equilateraltriangles.
1. Begin with step 9 of the DeZZbuilding block. Fold the quadrilateralin half along the diagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
11. An octahedron takes 12 units. 12. And an icosahedron takes 30units.
13. With 210 units, one can make adeltahedrified snub dodecahedron.However, the very shallow angles meansthat it doesn’t hold together very well.
Deltahedrally Elevated PolyhedraElevation is the result of erecting a pyramid on each face. If the resulting new faces are equilateraltriangles, then we can fold them from still another version of this unit that makes each face aseamless equilateral triangle.
1. Begin with step 10 of the DeZZbuilding block. Bring two pointstogether.
2. Fold the upper lefttriangle behind and thebottom triangle up.
3. Partially unfoldall of the foldsalong the strip.
4. One unit makesa portion of adouble pyramid.
5. 4 units makes anelevated tetrahedron,which resembles a caltrop.
6. The 12-unit elevatedoctahedron is also astellation of theoctahedron; Kepler calledit the Stella Octangula.
8. The elevated cube takes12 units, and resembles aslightly stubbier version ofthe origami model calledthe Jackstone.
9. The 30-unit elevateddodecahedron is a slightly bumpyball that is close to, but not exactly,a rhombic triacontahedron.
10. The 30-unit elevated icosahedronis considerably bumpier. It is closeto, but not exactly, a stellation of thedodecahedron.
11. The elevated cuboctahedrontakes 24 units.
13. And finally, 200 units will buildyou the elevated smallrhombicosidodecahedron.
14. If you make 3 units with allmountain folds, they can beassembled into the deltahedralequivalent of Takahama’s Jewel.
15 . Which is a deltahedrallyelevated trigonal dihedron, or,more simply, a tetrahedraldipyramid.
18. If a polyhedron is elevatedwith negative height, we call it“depressed.” You can folddepressed polyhedra by changingthe parity of some of the creaseslike this.
19 . This depressed dodecahedronrequires 30 units, like its elevatedkin.
20. Leaving out the middle creasegives a unit that has an interestingapplication...
21. If you elevate the triangles anddepress the pentagons of anicosidodecahedron, you get thisshape, which also happens to bethe logo of Wolfram | Alpha.
23. If you depress the trianglesand elevate the pentagons of theicosidodecahedron, you get anicosahedron with holes.
22. With the mountain fold backin place, we can make other mixedelevated/depressed polyhedra...
24. We can treat the cuboctahedronsimilarly. Elevated triangles, depressedsquares in a cuboctahedron.
25. Elevated squares, depressedtriangles in a cuboctahedron.
16 . Four units gives adeltahedrally elevated squaredihedron, or a square dipyramid,or simply, an octahedron.
17. And 5 such units gives apentagonal dipyramid.
26. And finally, to wrap up, going back to the originalelevated unit, 210 units give a deltahedrally-elevateddeltahedrified snub dodecahedron. (Yes, that’s double-deltahedrification!)
-
12. The elevated icosidodecahedrontakes 60 units. Leonardo da Vincidescribed (and named) this solid.
DeZZ UnitCopyright ©2012 by Robert J. Lang
This is actually several units in one: a Deltahedron Zig-Zag unit, which can be used to fold any deltahedron (anypolyhedron whose faces are equilateral triangles). A variation of the unit lets you fold a twisted-hole cube; anothervariation works for any deltahedrally elevated polyhedron; another variation folds a rhomboidal polyhedron that isthe Wolfram Alpha logo. The units draw upon concepts identified and explored by Bob Neale, Lewis Simon, andMitsunobu Sonobe, not to mention Tom Hull’s famous PHiZZ unit, which provides, as well, the rationale for thismodule’s name.
1. Begin with a square,colored side up. Fold inhalf vertically andunfold, making a pinchat the left.
2. Fold the bottom leftcorner to the mark youjust made, creasing aslightly as possible.
3. Fold and unfold alongan angle bisector,making a pinch along theright edge.
4. Fold the top left cornerdown to the creaseintersection.
black dot
hollow dot
arrow tiny
arrow small
arrow large
turn over
rotate cw
rotate ccw
view from here
repeat ll
repeat lr
repeat ul
repeat ur
right angle
cut here
3. Fold and unfold. Repeat behind.
4. Fold and unfold. Repeat behind. 5. Unfold so that the coloredside is visible.
6. The mountain creases here showthe folds that are used for thedeltahedron. Fold 3N/2 units for adeltahedron with N faces.
7. Here is how two units gotogether. The tab slips into thepocket on the back side, and thetwo points marked by dots cometogether.
8. The third unit goes insideone pocket and outside onetab.
9. The dots are the corners of atriangular face. Keep adding unitsto create any deltahedron.
10. Here is a tetrahedron, from 6units.
5. Turn the paper over. 6. Fold the top foldededge down to the rawbottom edge.
7. Fold the bottom foldededge (but not the raw edgebehind it) up to the top andunfold.
8. Fold the top left corner andthe bottom right corner to thecrease you just made.
9. Here’s the building block. 10. Here’s the other side.
Plain Twisted-Hole CubeThis structure is similar to Lewis Simon’s many twist-hole cubes, but uses the assembly technique of RobertNeale’s dodecahedron.
1. Begin with step 9 of the DeZZbuilding block. Fold thequadrilateral in half along thediagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
3. Join three units at a corner bysliding tabs into pockets.
4. The finished cube.
DeltahedraThis unit can be used to make anypolyhedron whose faces are equilateraltriangles.
1. Begin with step 9 of the DeZZbuilding block. Fold the quadrilateralin half along the diagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
11. An octahedron takes 12 units. 12. And an icosahedron takes 30units.
13. With 210 units, one can make adeltahedrified snub dodecahedron.However, the very shallow angles meansthat it doesn’t hold together very well.
Deltahedrally Elevated PolyhedraElevation is the result of erecting a pyramid on each face. If the resulting new faces are equilateraltriangles, then we can fold them from still another version of this unit that makes each face aseamless equilateral triangle.
1. Begin with step 10 of the DeZZbuilding block. Bring two pointstogether.
2. Fold the upper lefttriangle behind and thebottom triangle up.
3. Partially unfoldall of the foldsalong the strip.
4. One unit makesa portion of adouble pyramid.
5. 4 units makes anelevated tetrahedron,which resembles a caltrop.
6. The 12-unit elevatedoctahedron is also astellation of theoctahedron; Kepler calledit the Stella Octangula.
8. The elevated cube takes12 units, and resembles aslightly stubbier version ofthe origami model calledthe Jackstone.
9. The 30-unit elevateddodecahedron is a slightly bumpyball that is close to, but not exactly,a rhombic triacontahedron.
10. The 30-unit elevated icosahedronis considerably bumpier. It is closeto, but not exactly, a stellation of thedodecahedron.
11. The elevated cuboctahedrontakes 24 units.
13. And finally, 200 units will buildyou the elevated smallrhombicosidodecahedron.
14. If you make 3 units with allmountain folds, they can beassembled into the deltahedralequivalent of Takahama’s Jewel.
15 . Which is a deltahedrallyelevated trigonal dihedron, or,more simply, a tetrahedraldipyramid.
18. If a polyhedron is elevatedwith negative height, we call it“depressed.” You can folddepressed polyhedra by changingthe parity of some of the creaseslike this.
19 . This depressed dodecahedronrequires 30 units, like its elevatedkin.
20. Leaving out the middle creasegives a unit that has an interestingapplication...
21. If you elevate the triangles anddepress the pentagons of anicosidodecahedron, you get thisshape, which also happens to bethe logo of Wolfram | Alpha.
23. If you depress the trianglesand elevate the pentagons of theicosidodecahedron, you get anicosahedron with holes.
22. With the mountain fold backin place, we can make other mixedelevated/depressed polyhedra...
24. We can treat the cuboctahedronsimilarly. Elevated triangles, depressedsquares in a cuboctahedron.
25. Elevated squares, depressedtriangles in a cuboctahedron.
16 . Four units gives adeltahedrally elevated squaredihedron, or a square dipyramid,or simply, an octahedron.
17. And 5 such units gives apentagonal dipyramid.
26. And finally, to wrap up, going back to the originalelevated unit, 210 units give a deltahedrally-elevateddeltahedrified snub dodecahedron. (Yes, that’s double-deltahedrification!)
-
12. The elevated icosidodecahedrontakes 60 units. Leonardo da Vincidescribed (and named) this solid.
DeZZ UnitCopyright ©2012 by Robert J. Lang
This is actually several units in one: a Deltahedron Zig-Zag unit, which can be used to fold any deltahedron (anypolyhedron whose faces are equilateral triangles). A variation of the unit lets you fold a twisted-hole cube; anothervariation works for any deltahedrally elevated polyhedron; another variation folds a rhomboidal polyhedron that isthe Wolfram Alpha logo. The units draw upon concepts identified and explored by Bob Neale, Lewis Simon, andMitsunobu Sonobe, not to mention Tom Hull’s famous PHiZZ unit, which provides, as well, the rationale for thismodule’s name.
1. Begin with a square,colored side up. Fold inhalf vertically andunfold, making a pinchat the left.
2. Fold the bottom leftcorner to the mark youjust made, creasing aslightly as possible.
3. Fold and unfold alongan angle bisector,making a pinch along theright edge.
4. Fold the top left cornerdown to the creaseintersection.
black dot
hollow dot
arrow tiny
arrow small
arrow large
turn over
rotate cw
rotate ccw
view from here
repeat ll
repeat lr
repeat ul
repeat ur
right angle
cut here
3. Fold and unfold. Repeat behind.
4. Fold and unfold. Repeat behind. 5. Unfold so that the coloredside is visible.
6. The mountain creases here showthe folds that are used for thedeltahedron. Fold 3N/2 units for adeltahedron with N faces.
7. Here is how two units gotogether. The tab slips into thepocket on the back side, and thetwo points marked by dots cometogether.
8. The third unit goes insideone pocket and outside onetab.
9. The dots are the corners of atriangular face. Keep adding unitsto create any deltahedron.
10. Here is a tetrahedron, from 6units.
5. Turn the paper over. 6. Fold the top foldededge down to the rawbottom edge.
7. Fold the bottom foldededge (but not the raw edgebehind it) up to the top andunfold.
8. Fold the top left corner andthe bottom right corner to thecrease you just made.
9. Here’s the building block. 10. Here’s the other side.
Plain Twisted-Hole CubeThis structure is similar to Lewis Simon’s many twist-hole cubes, but uses the assembly technique of RobertNeale’s dodecahedron.
1. Begin with step 9 of the DeZZbuilding block. Fold thequadrilateral in half along thediagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
3. Join three units at a corner bysliding tabs into pockets.
4. The finished cube.
DeltahedraThis unit can be used to make anypolyhedron whose faces are equilateraltriangles.
1. Begin with step 9 of the DeZZbuilding block. Fold the quadrilateralin half along the diagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
11. An octahedron takes 12 units. 12. And an icosahedron takes 30units.
13. With 210 units, one can make adeltahedrified snub dodecahedron.However, the very shallow angles meansthat it doesn’t hold together very well.
Deltahedrally Elevated PolyhedraElevation is the result of erecting a pyramid on each face. If the resulting new faces are equilateraltriangles, then we can fold them from still another version of this unit that makes each face aseamless equilateral triangle.
1. Begin with step 10 of the DeZZbuilding block. Bring two pointstogether.
2. Fold the upper lefttriangle behind and thebottom triangle up.
3. Partially unfoldall of the foldsalong the strip.
4. One unit makesa portion of adouble pyramid.
5. 4 units makes anelevated tetrahedron,which resembles a caltrop.
6. The 12-unit elevatedoctahedron is also astellation of theoctahedron; Kepler calledit the Stella Octangula.
8. The elevated cube takes12 units, and resembles aslightly stubbier version ofthe origami model calledthe Jackstone.
9. The 30-unit elevateddodecahedron is a slightly bumpyball that is close to, but not exactly,a rhombic triacontahedron.
10. The 30-unit elevated icosahedronis considerably bumpier. It is closeto, but not exactly, a stellation of thedodecahedron.
11. The elevated cuboctahedrontakes 24 units.
13. And finally, 200 units will buildyou the elevated smallrhombicosidodecahedron.
14. If you make 3 units with allmountain folds, they can beassembled into the deltahedralequivalent of Takahama’s Jewel.
15 . Which is a deltahedrallyelevated trigonal dihedron, or,more simply, a tetrahedraldipyramid.
18. If a polyhedron is elevatedwith negative height, we call it“depressed.” You can folddepressed polyhedra by changingthe parity of some of the creaseslike this.
19 . This depressed dodecahedronrequires 30 units, like its elevatedkin.
20. Leaving out the middle creasegives a unit that has an interestingapplication...
21. If you elevate the triangles anddepress the pentagons of anicosidodecahedron, you get thisshape, which also happens to bethe logo of Wolfram | Alpha.
23. If you depress the trianglesand elevate the pentagons of theicosidodecahedron, you get anicosahedron with holes.
22. With the mountain fold backin place, we can make other mixedelevated/depressed polyhedra...
24. We can treat the cuboctahedronsimilarly. Elevated triangles, depressedsquares in a cuboctahedron.
25. Elevated squares, depressedtriangles in a cuboctahedron.
16 . Four units gives adeltahedrally elevated squaredihedron, or a square dipyramid,or simply, an octahedron.
17. And 5 such units gives apentagonal dipyramid.
26. And finally, to wrap up, going back to the originalelevated unit, 210 units give a deltahedrally-elevateddeltahedrified snub dodecahedron. (Yes, that’s double-deltahedrification!)
-
12. The elevated icosidodecahedrontakes 60 units. Leonardo da Vincidescribed (and named) this solid.
DeZZ UnitCopyright ©2012 by Robert J. Lang
This is actually several units in one: a Deltahedron Zig-Zag unit, which can be used to fold any deltahedron (anypolyhedron whose faces are equilateral triangles). A variation of the unit lets you fold a twisted-hole cube; anothervariation works for any deltahedrally elevated polyhedron; another variation folds a rhomboidal polyhedron that isthe Wolfram Alpha logo. The units draw upon concepts identified and explored by Bob Neale, Lewis Simon, andMitsunobu Sonobe, not to mention Tom Hull’s famous PHiZZ unit, which provides, as well, the rationale for thismodule’s name.
1. Begin with a square,colored side up. Fold inhalf vertically andunfold, making a pinchat the left.
2. Fold the bottom leftcorner to the mark youjust made, creasing aslightly as possible.
3. Fold and unfold alongan angle bisector,making a pinch along theright edge.
4. Fold the top left cornerdown to the creaseintersection.
black dot
hollow dot
arrow tiny
arrow small
arrow large
turn over
rotate cw
rotate ccw
view from here
repeat ll
repeat lr
repeat ul
repeat ur
right angle
cut here
3. Fold and unfold. Repeat behind.
4. Fold and unfold. Repeat behind. 5. Unfold so that the coloredside is visible.
6. The mountain creases here showthe folds that are used for thedeltahedron. Fold 3N/2 units for adeltahedron with N faces.
7. Here is how two units gotogether. The tab slips into thepocket on the back side, and thetwo points marked by dots cometogether.
8. The third unit goes insideone pocket and outside onetab.
9. The dots are the corners of atriangular face. Keep adding unitsto create any deltahedron.
10. Here is a tetrahedron, from 6units.
5. Turn the paper over. 6. Fold the top foldededge down to the rawbottom edge.
7. Fold the bottom foldededge (but not the raw edgebehind it) up to the top andunfold.
8. Fold the top left corner andthe bottom right corner to thecrease you just made.
9. Here’s the building block. 10. Here’s the other side.
Plain Twisted-Hole CubeThis structure is similar to Lewis Simon’s many twist-hole cubes, but uses the assembly technique of RobertNeale’s dodecahedron.
1. Begin with step 9 of the DeZZbuilding block. Fold thequadrilateral in half along thediagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
3. Join three units at a corner bysliding tabs into pockets.
4. The finished cube.
DeltahedraThis unit can be used to make anypolyhedron whose faces are equilateraltriangles.
1. Begin with step 9 of the DeZZbuilding block. Fold the quadrilateralin half along the diagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
11. An octahedron takes 12 units. 12. And an icosahedron takes 30units.
13. With 210 units, one can make adeltahedrified snub dodecahedron.However, the very shallow angles meansthat it doesn’t hold together very well.
Deltahedrally Elevated PolyhedraElevation is the result of erecting a pyramid on each face. If the resulting new faces are equilateraltriangles, then we can fold them from still another version of this unit that makes each face aseamless equilateral triangle.
1. Begin with step 10 of the DeZZbuilding block. Bring two pointstogether.
2. Fold the upper lefttriangle behind and thebottom triangle up.
3. Partially unfoldall of the foldsalong the strip.
4. One unit makesa portion of adouble pyramid.
5. 4 units makes anelevated tetrahedron,which resembles a caltrop.
6. The 12-unit elevatedoctahedron is also astellation of theoctahedron; Kepler calledit the Stella Octangula.
8. The elevated cube takes12 units, and resembles aslightly stubbier version ofthe origami model calledthe Jackstone.
9. The 30-unit elevateddodecahedron is a slightly bumpyball that is close to, but not exactly,a rhombic triacontahedron.
10. The 30-unit elevated icosahedronis considerably bumpier. It is closeto, but not exactly, a stellation of thedodecahedron.
11. The elevated cuboctahedrontakes 24 units.
13. And finally, 200 units will buildyou the elevated smallrhombicosidodecahedron.
14. If you make 3 units with allmountain folds, they can beassembled into the deltahedralequivalent of Takahama’s Jewel.
15 . Which is a deltahedrallyelevated trigonal dihedron, or,more simply, a tetrahedraldipyramid.
18. If a polyhedron is elevatedwith negative height, we call it“depressed.” You can folddepressed polyhedra by changingthe parity of some of the creaseslike this.
19 . This depressed dodecahedronrequires 30 units, like its elevatedkin.
20. Leaving out the middle creasegives a unit that has an interestingapplication...
21. If you elevate the triangles anddepress the pentagons of anicosidodecahedron, you get thisshape, which also happens to bethe logo of Wolfram | Alpha.
23. If you depress the trianglesand elevate the pentagons of theicosidodecahedron, you get anicosahedron with holes.
22. With the mountain fold backin place, we can make other mixedelevated/depressed polyhedra...
24. We can treat the cuboctahedronsimilarly. Elevated triangles, depressedsquares in a cuboctahedron.
25. Elevated squares, depressedtriangles in a cuboctahedron.
16 . Four units gives adeltahedrally elevated squaredihedron, or a square dipyramid,or simply, an octahedron.
17. And 5 such units gives apentagonal dipyramid.
26. And finally, to wrap up, going back to the originalelevated unit, 210 units give a deltahedrally-elevateddeltahedrified snub dodecahedron. (Yes, that’s double-deltahedrification!)
-
12. The elevated icosidodecahedrontakes 60 units. Leonardo da Vincidescribed (and named) this solid.
DeZZ UnitCopyright ©2012 by Robert J. Lang
This is actually several units in one: a Deltahedron Zig-Zag unit, which can be used to fold any deltahedron (anypolyhedron whose faces are equilateral triangles). A variation of the unit lets you fold a twisted-hole cube; anothervariation works for any deltahedrally elevated polyhedron; another variation folds a rhomboidal polyhedron that isthe Wolfram Alpha logo. The units draw upon concepts identified and explored by Bob Neale, Lewis Simon, andMitsunobu Sonobe, not to mention Tom Hull’s famous PHiZZ unit, which provides, as well, the rationale for thismodule’s name.
1. Begin with a square,colored side up. Fold inhalf vertically andunfold, making a pinchat the left.
2. Fold the bottom leftcorner to the mark youjust made, creasing aslightly as possible.
3. Fold and unfold alongan angle bisector,making a pinch along theright edge.
4. Fold the top left cornerdown to the creaseintersection.
black dot
hollow dot
arrow tiny
arrow small
arrow large
turn over
rotate cw
rotate ccw
view from here
repeat ll
repeat lr
repeat ul
repeat ur
right angle
cut here
3. Fold and unfold. Repeat behind.
4. Fold and unfold. Repeat behind. 5. Unfold so that the coloredside is visible.
6. The mountain creases here showthe folds that are used for thedeltahedron. Fold 3N/2 units for adeltahedron with N faces.
7. Here is how two units gotogether. The tab slips into thepocket on the back side, and thetwo points marked by dots cometogether.
8. The third unit goes insideone pocket and outside onetab.
9. The dots are the corners of atriangular face. Keep adding unitsto create any deltahedron.
10. Here is a tetrahedron, from 6units.
5. Turn the paper over. 6. Fold the top foldededge down to the rawbottom edge.
7. Fold the bottom foldededge (but not the raw edgebehind it) up to the top andunfold.
8. Fold the top left corner andthe bottom right corner to thecrease you just made.
9. Here’s the building block. 10. Here’s the other side.
Plain Twisted-Hole CubeThis structure is similar to Lewis Simon’s many twist-hole cubes, but uses the assembly technique of RobertNeale’s dodecahedron.
1. Begin with step 9 of the DeZZbuilding block. Fold thequadrilateral in half along thediagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
3. Join three units at a corner bysliding tabs into pockets.
4. The finished cube.
DeltahedraThis unit can be used to make anypolyhedron whose faces are equilateraltriangles.
1. Begin with step 9 of the DeZZbuilding block. Fold the quadrilateralin half along the diagonal.
2. Unfold all the creases to 90°dhiedral angles. Make 12 units.
11. An octahedron takes 12 units. 12. And an icosahedron takes 30units.
13. With 210 units, one can make adeltahedrified snub dodecahedron.However, the very shallow angles meansthat it doesn’t hold together very well.
Deltahedrally Elevated PolyhedraElevation is the result of erecting a pyramid on each face. If the resulting new faces are equilateraltriangles, then we can fold them from still another version of this unit that makes each face aseamless equilateral triangle.
1. Begin with step 10 of the DeZZbuilding block. Bring two pointstogether.
2. Fold the upper lefttriangle behind and thebottom triangle up.
3. Partially unfoldall of the foldsalong the strip.
4. One unit makesa portion of adouble pyramid.
5. 4 units makes anelevated tetrahedron,which resembles a caltrop.
6. The 12-unit elevatedoctahedron is also astellation of theoctahedron; Kepler calledit the Stella Octangula.
8. The elevated cube takes12 units, and resembles aslightly stubbier version ofthe origami model calledthe Jackstone.
9. The 30-unit elevateddodecahedron is a slightly bumpyball that is close to, but not exactly,a rhombic triacontahedron.
10. The 30-unit elevated icosahedronis considerably bumpier. It is closeto, but not exactly, a stellation of thedodecahedron.
11. The elevated cuboctahedrontakes 24 units.
13. And finally, 200 units will buildyou the elevated smallrhombicosidodecahedron.
14. If you make 3 units with allmountain folds, they can beassembled into the deltahedralequivalent of Takahama’s Jewel.
15 . Which is a deltahedrallyelevated trigonal dihedron, or,more simply, a tetrahedraldipyramid.
18. If a polyhedron is elevatedwith negative height, we call it“depressed.” You can folddepressed polyhedra by changingthe parity of some of the creaseslike this.
19 . This depressed dodecahedronrequires 30 units, like its elevatedkin.
20. Leaving out the middle creasegives a unit that has an interestingapplication...
21. If you elevate the triangles anddepress the pentagons of anicosidodecahedron, you get thisshape, which also happens to bethe logo of Wolfram | Alpha.
23. If you depress the trianglesand elevate the pentagons of theicosidodecahedron, you get anicosahedron with holes.
22. With the mountain fold backin place, we can make other mixedelevated/depressed polyhedra...
24. We can treat the cuboctahedronsimilarly. Elevated triangles, depressedsquares in a cuboctahedron.
25. Elevated squares, depressedtriangles in a cuboctahedron.
16 . Four units gives adeltahedrally elevated squaredihedron, or a square dipyramid,or simply, an octahedron.
17. And 5 such units gives apentagonal dipyramid.
26. And finally, to wrap up, going back to the originalelevated unit, 210 units give a deltahedrally-elevateddeltahedrified snub dodecahedron. (Yes, that’s double-deltahedrification!)