Semi Cubes
www.walser-h-m.ch/hans/Vortraege Hans Walser
unit cube half volume
12
3 ≈ 0.79
Not possible with ruler and compass
12
2 ≈ 0.71
half edgelength
12
1 = 12 1
half surface Semi cubes
Bisecting the volume
Counterpart point-symmetric and mirror-inverted Bisecting the volume
Bisecting the volume
Hyperbolic paraboloid
Monkey-saddle
z = x3 − 3xy2
Cube on a vertex
Intersection with monkey-saddle
Counterpart point-symmetric and mirror-inverted Intersection with monkey-saddle
Intersection with monkey-saddle
Counterpart point-symmetric and mirror-inverted
Intersection with monkey-saddle
Counterpart point-symmetric and mirror-inverted
Intersection with monkey-saddle
Counterpart point-symmetric and mirror-inverted
Intersection with monkey-saddle
Counterpart point-symmetric and mirror-inverted
Intersection with monkey-saddle
Counterpart point-symmetric and mirror-inverted
Fitting in the coordinate-system
Semi-cube
Fitting in the coordinate-system
Semi-cube
Reflecting at the coordinate-planes
Semi-cube
Reflecting at the coordinate-planes
Semi-cube
Twelfepede - dodecapus
Reflecting at the coordinate-planes
Spacefiller
Twelfepede - dodecapus
Spacefiller
Spacefiller
Spacefiller
Spacefiller
Spacefiller
Spacefiller
Spacefiller
Collision Spacefiller
Spacefiller
Spacefiller
Nonempty intersection Spacefiller
Spacefiller
Fits
But you can‘t fit it Spacefiller
Newstart: Far away Spacefiller
Simultaneous fitting Spacefiller
Simultaneous fitting Spacefiller
Simultaneous fitting Spacefiller
Simultaneous fitting Spacefiller
Fits
Simultaneous fitting Spacefiller
Simultaneous fitting Spacefiller
Simultaneous fitting Spacefiller
Half-cube
Half-cube
Half-cube
Half-cube
Half-cube
3d-cross
Filling the space
Joining together without problems
Fitting without problems Filling the space
Filling the space
Fitting without problems
Filling the space
Fitting without problems
Filling the space
Fitting without problems
Filling the space
Fitting without problems
Filling the space
Fitting without problems
Filling the space
Fitting without problems
Filling the space
Fitting without problems
Half-cube
Half-cube
Half-cube
Filling the space
Joining together without problems
Fits. Fitting together impossible Filling the space
Filling the space
Fits. Fitting together impossible
Filling the space
Fits. Fitting together impossible
Filling the space
Fits. Fitting together impossible
Filling the space
Fits. Fitting together impossible
Filling the space
Fits. Fitting together impossible
Filling the space
Fits. Fitting together impossible
Gap filler
The implant fits. Fitting it into the gap impossible
Filling the plane
Fits. Fitting together in the plane impossible
Nuts with different symmetries
How can we see the kernel without cracking the nut?
Two fold Three fold Four fold
Analogue 2d problem
How can we see the kernel without cracking the nut?
Analogue 2d problem Embedding in 3d space
How can we see the kernel without cracking the nut?
How can we see the kernel without cracking the nut?
Analogue 2d problem Embedding in 3d space
Remove the kernel
3d Origami
Not like this 3d Origami
3d Origami
Just 2d Origami
2d Origami
Folding a vertex on the opposite vertex 2d Origami
2d Origami
2d Origami works only in 3d space
2d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
“Folding” in 4d hyperspace 3d Origami
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
3d Origami
Folding a vertex on the opposite vertex
2d Origami
Folding the midpoints of the edges on the center
2d Origami
Folding the midpoints of the edges on the center
2d Origami
Folding the midpoints of the edges on the center
2d Origami
Folding the midpoints of the edges on the center
2d Origami
Folding the midpoints of the edges on the center
2d Origami
Folding the vertices onto the center
2d Origami
Folding the vertices onto the center
2d Origami
Folding the vertices onto the center
2d Origami
Folding the vertices onto the center
2d Origami
Folding the vertices onto the center
3d Origami
“Folding” in 4d hyperspace
3d Origami
Folding the midpoints of the faces into the center
3d Origami
Folding the midpoints of the faces into the center
3d Origami
Folding the midpoints of the faces into the center
3d Origami
Folding the midpoints of the faces into the center
3d Origami
Folding the midpoints of the faces into the center
3d Origami
Folding the midpoints of the faces into the center
3d Origami
One-eighth of the volume
3d Origami
“Folding” in 4d hyperspace
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
Origami im Raum
Kirchturm bei Krefeld
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
3d Origami
Folding the midpoints of the edges into the center
Rhombic dodecahedron. Quarter of the volume 3d Origami
Rhombic dodecahedron as space filler
View from above. How many colors?
Rhombic dodecahedron as space filler
How many colors in space?
How many colors in space?
How many colors in space?
How many colors in space?
How many colors in space?
One, two, three, four, five at least
One, two, three, four, five at least
One, two, three, four, five at least
One, two, three, four, five at least
One, two, three, four, five at least
Rhombic dodecahedron as space filler
Chessboard
Chessboard
Second chessboard
Rhombic dodecahedron as space filler
Rhombic dodecahedron as space filler
Rhombic dodecahedron as space filler
Rhombic dodecahedron as space filler
Rhombic dodecahedron as space filler
Rhombic dodecahedron as space filler
3d Origami
“Folding” in 4d hyperspace
3d Origami
Folding the vertices into the center
3d Origami
Folding the vertices into the center
3d Origami
Folding the vertices into the center
3d Origami
Folding the vertices into the center
3d Origami
Folding the vertices into the center
3d Origami
Folding the vertices into the center
3d Origami
Folding the vertices into the center
3d Origami
Folding the vertices into the center
3d Origami
Folding the vertices into the center
Truncated Octahedron. Half of the volume 3d Origami
Truncated Octahedron. Half of the volume
Body-centered cubic system
Body-centered cubic system
Thank you
Body-centered cubic system
