# Dividing a sphere into twelve "identical" shapes.

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Imatfaal,

Right you are. I read the words but didn't see what you meant, 'till the other day.

But then again, I am 60. I meet new people all the time. And have new ideas all the time.

Regards, TAR

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Imatfaal,

Well wait. I did understand what you were saying in the link about the cubic octahedron being the dual of the rhombic dodecahedron.

What I noticed the other day was that a pure string colored octahedron, with eight triangular faces was inside the figure, with each face exactly oriented to a vertex of the purple cube. And vice-a-versa. Each face of the cube was exactly oriented to a vertex of the octahedron.

The octahedron has eight faces and 6 vertices. The cube has 6 faces and eight vertices. And each has 12 edges. The middle of an edge of the cube is exactly on the same ray from the center of the complex as the center of an edge of the octahedron (although the edges do not intersect at a point, just intersect the same ray from the center of the complex, which incidently is on the same line as the ray projecting 180 degrees the other way, through the center points of the opposite edges of both the cube and the octahedron.)

Although ultimately, the meaning of cubic octahedron and the meaning of a cube and an octahedron, might be the same after inspection, there are some distinctions and some interesting relatiions intertwined in the distinctions.

Most interesting to me about this, is the fact that the diagonals of the diamonds, when inspecting the spherical rhomic dodecahedron do no touch, and the edges of the cube are "outside" the edges of the octahedron.

Regards, TAR

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Don't know what it means yet, but made an octahedron from clay, stuck a toothpick in each apex and in the center of each face, for reference, and cut out the diamond shapes by making cuts between the toothpicks. Exactly twelve, with exactly the same internal angles as cutting the sphere and the cube into the twelve sections. The "edge" was exactly 90 degrees off from the cube segments but you could switch one segment from the cube of the same far corner to corner size as a thusly cut up octahedron of the same far apex to apex size. Have not figured out exactly how that ties back to the figure (the spherical rhombic dodecahedron) but I thought it worth mentioning that the octahedron was exactly setup anglewise to be sectioned in the same twelve internal angle sections. And of course the twelve edges, and the being a dual of the cube, helps.

Edited by tar

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Go to bed Tar! I know it is fascinating but a man needs to sleep. Unless you are working nights in which case welcome to Greenwich Mean Time. Will try and figure what you are getting - but I am working today and gonna log of soon

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Imatfaal,

Talk to you later. Once I figure out what I am working on.

Regards, TAR

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Imatfaal,

I think what I am working on, is a directional, geometrical understanding of the universe. A scheme that gives permanent structure and relationship to all elements of the cosmos.

The division of a sphere or a cube or an octahedron into the same 12 angular spaces can be like wise projected outward from a center point indefinitely and can be "done" starting from any point in space and time.

Such a three dimensional scheme is already in place with our degrees of ascension and declination, but it uses the observer's and the Earth's reference points as the basis of measurement, which does not "nail down" the scheme and tie it to the greater universe.

I have wanted, from my late teens, to be able to know, whether day or night, which direction I was heading in, in reference to the center of the galaxy. To be able to close my eyes and imagine the Earth's spin around it's axis, its revolution around the Sun, and the Sun's tilt and progression around the center of the galaxy...all at once. In one "picture", based on where it was I was sitting/standing/lying on the surface of the Earth.

...The other night, I was looking at the moon and imagining its progression Eastward around the Earth as it was heading to set in the West, and linked the Earth/Moon combine in its progression around the Sun, to a "picture" of the position and tilt of the Earth and its progression around the Sun, to such a picture I had formulated around the same time of year 30 some years ago in Germany.

I think my crazyman drawing of the 12 diamond shapes onto the world, from my deck over the summer, has increased my geometrical understanding of the place... and brought me one step closer to being able to see the whole arrangement, at once.

Regards, TAR

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Been trying to come up with a way to depict any direction using the 12 segments (spherical rhombic dodecahedron shapes) and finally did yesterday, after making a realization the night before.

I had a clay ball with long toothpicks at tje eight corners of the cube(blue wirenuts in previous picure) and short tootpicks at the points of the octahedron(orange wirenuts). I scored the ball between the toothpicks to make the 12 spherical rhombic dodecahedron shapes. I put a large penhole in the center of each shape. Then I noticed (again) that the holes lined up in the 4 hexagonal planes we talked about. I placed small penholes every 10 degrees and looked at the relationship the holes had to the grid when you divide the shape in quarters by drawing a line parallel and inbetween the sides of the shape.

I noticed the angle was off, but it was off consistently, with the diamonds facing one way then the other.

If i made 20 degree marks in the hexagonal plane I could use those marks to draw my parallel grid.

Each diamond was thusly dividable into "degrees" "minutes" "seconds" and fractions of seconds if you wished. Any point on the sphere could be named, exactly and reproducably, as long as you had a naming convention, which I developed yesterday, by starting with the center of an equatorial diamond and going up and to the right, around the hexagon, and then the other hexagonal plane that goes through that point, up and to the left and all the way round. 6 60degree sections adding up to 360 degrees. Then I moved over on diamond on the equator and repeated the process. Each of the four planes has its own letter/color, and I used a Capital Letter to signify having the degrees increase upward and a little letter having the degrees increase downward. Thusly each diamond has its own name and numbering convention and is related to actual degrees.

The lines (degrees) on my grid are a little shorter than the actual degrees so there are more square TAR degrees than actual square degrees on the sphere. TAR degrees come to 43200 square degrees (60x60x12) Actual square degrees in sphere are 41253 (approx) but the conversion from TAR square degrees to actual square degrees is regular and acurate. 1.0471 times 41253 is 43200 and of course the reciprical .9549 times 43200 is 41253. 1.0471 is 1/3 Pi. so the scheme is usable and convertable. And a whole lot easier than the figuring you have to do using spherical coordinants.

Regards, TAR

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Scheme drawn on pingpong ball. Same ball, five different shots, sort of following the red hexagonal plane around.

Edited by tar

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Here is more carefully drawn pingpong ball, with the lines depicting 10 degree divisions as opposed to the 15 degrees shown just before. Also arranged in same pattern as previous drawing of the 12 sections laid out to be seen at once, for better visualization.

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Interesting observation this morning. The cube octahedron that started this investigation popped back up when describing the four hexagonal planes. In the figure here (rotated 90 degrees CW to orient to diamond number one, the RY diamond conceptually) the toothpicks are normal to each of the four hexagonal planes, describing four axis. The toothpicks are at the eight "threepoints" of the spherical rhombic dodecahedron. This configuration (my observation) is exactly the same orientation as if you stuck a toothpick in each of the eight corners of a cube. Holding one of said axis and spinning figure your center passes through each of the six balls that are positioned at the center of each of the diamonds. These correspond to the center of the six edges of a cube that are not the six edges that your axis toothpicks are stuck into the ends of. Exactly. Interesting figure.

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Might have a problem.

The lines that divide the diamond into TAR degree diamonds may not be consistent.

We already know that the two divisions per 1/12 of a sphere diamonds at the three points are shield shaped and not diamond shape. This was OK in my book, because as you divided into minutes and seconds the shield shape would be smaller and smaller and less of a percentage of the total diamond and would probably not interfer to badly with most mapping applications of the figure.

However, there might be a difference in the size of the division diamonds, comparing central divisions with toward the edges divisions. Not sure yet, since this figure is surprising, but according to the trial pictured in this post, the dividing lines are great circles around the sphere. As you can see the dividing lines if extended around the sphere, intersect at the poles, form the same exact divisions on the diametrically opposed diamond (not shown). Looking at the spacing between these great circle lines, it reduces to 0 at the poles. This would seem to mean that the spacing would increase to maximum exactly half way between the poles, which would correspond to the center of the 1/12 diamond outlined with toothpick holes. This in turn would mean that the spacing closer to the diamond edges would be less, making the division diamonds, not consistent in size. Not sure yet, because the divisions seem regular, and propotional and the figure is based on the cube octahedron which has those "vector equilibrium" characteristics, but I am doing some thinking and trials and measurements on this, to determine the consistency or inconsistency of the size of the division diamonds.

Regards, TAR

Actually I misspoke. They are pen tip holes, not toothpick holes. The lines a drawn by moving a steak knife against the sphere in its own track, without a cutting motion attempting to keep the knife blade normal to the center of the sphere. The hexagona plane lines are made laying a toothpick against the sphere, in its own track.

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I don't think we can have 12 identical Sections comprising of the entire Solid Sphere with sphere's surface section cut to look like Square or higher polygons.

Only identical Spherical Triangles are possible which is quite obvious.

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I don't think we can have 12 identical Sections comprising of the entire Solid Sphere with sphere's surface section cut to look like Square or higher polygons.

Only identical Spherical Triangles are possible which is quite obvious.

Well pentagons would clearly work - imagine a bloated dodecahdron.

And if you read through the thread you will see that Tar has shown another version - it has also been modelled by a few members

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Yes, the divisions here are not meant to be Platonic solids with polyhedral faces. The divisions are actually drawn on the surface of a sphere. The geometry of the surface of a sphere is very interesting, as I discover more and more interesting characteristics of the spherical rhombic dodecahedreon, that Janus so nicely rendered for us, with the twelve identical solid sections, and all. (I am calling that solid shape you get when you cut along the lines of the spherical rhombic dodecadron to the center of the sphere, the Janus if it has not yet been named.)

I am less concerned now that the diamond divisions are of inconsistent size. The line where the spacing between the great circle lines on opposite sides of the diamond is the greatest is the hexagonal plane line. These lines are actually skewed to the diamond in such a way as to give width to the edge divisions as well as the more central divisions, and the divisions are, after all, direct proportions of the four diamond border lines, which we have already determined, are of equal length.

Found out a tentative answer to my earlier question of whether the length of the edge of one of the twelve sections was 1/6th of a circumference or was actually exactly a radius. While those two possibilities might not be the only ones, it appears, by crude measurement, that the lines are each a radius long.

Using an engineering ruler with 60 divisions per inch I walked the ruler around the great circle of one of my spheres and found it to be about 370 divisions. Roughly measuring the diameter, holding the ruler behind the sphere it looked around 118 divisions.

Dividing the 370 by Pi would get me a radius of 58.9 or so, dividing the circumference by 6 would be about 61.6. So I drew out my diamonds and measured the length of the 24 lines, adding them up and divided by 24 and got 58.58. This was closer to the 58.9 then to the 61.6, so I am proceeding as if the length of the lines is exactly one radius, until I prove, or it is proved otherwise.

Subsequently I cut toothpicks into 118 division lengths and found I could lay out the figure (the spherical rhombic dodecahedron) laying a 118 toothpick(one D) down in its on track, making a cross with another 118 and repeating this procedure the six places on the figure where the 4-points are. Works out exactly. The six places being the North and South Poles, and the four equatorial cross points described earlier.

Even more interesting was this tetrahedral figure, shown with just one toothpick at each of the six cross points.

Regards, TAR

The figure was made by placing on 118 D toothpick on each of the cross points and cutting off the four pieces of the sphere that stuck out beyond the subsequent tetrahedron, and then turning each of the toothpicks 90 degrees to represent the other diameter at the cross. Interestingly enough, unless I am wrong, it is exactly one diameter, along the surface of the sphere from one vertex of this tetrahedron like figure to the next vertex.

You will notice the six toothpicks arranged like this, orthogonically to the missing six form a dual tetrahedron to the one the other missing 6 would form.

It appears that the diameter of the sphere thusly laid out on surface, exactly meters out the surface into the spherical rhomic dodecahedron.

Neat.

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Taking the tetrahedral figure, with the edges being exactly two r long, and rolling it along the edges, you can describe a perfect hexagon on a 2D surface. Notice that 6 balls, placed exactly on the vertices of this hexagon, will fit exactly, as securely as 6 pennies around a central penny. The 118D toothpick lengths were placed on this track to prove that the toothpick lengths are 2r or one D, each. The spherical rhombic dodecahedron can be completely described on a sphere with 6 orthogonal crosses made with toothpicks of this diameter length.

Quite interesting that three Janus sections, make a 1/4 of a sphere, and each edge of one of these quarter sphere shapes, along the surface of the sphere, is exactly on diameter in length.

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Well pentagons would clearly work - imagine a bloated dodecahdron.

And if you read through the thread you will see that Tar has shown another version - it has also been modelled by a few members

imatfaal :

Thank you. Yes, I see the possibilities now more clearly.

Both Square and Pentagon can be used to have 12 identical Sections.

12 Triangles can also be used with identical but not equilateral shapes. This is the only possibility where the sphere is cut into 4 through Cuts ie. One horizontal and three Vertical Cuts every time passing through the center of the sphere splitting it into two.

Yes, the divisions here are not meant to be Platonic solids with polyhedral faces. The divisions are actually drawn on the surface of a sphere. The geometry of the surface of a sphere is very interesting, as I discover more and more interesting characteristics of the spherical rhombic dodecahedreon, that Janus so nicely rendered for us, with the twelve identical solid sections, and all. (I am calling that solid shape you get when you cut along the lines of the spherical rhombic dodecadron to the center of the sphere, the Janus if it has not yet been named.)

tar :

Thank you, yes I agree

Janus & tar :

Thank you for the good illustrations !

Edited by Commander

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Here are two unillustrated methods to make the pattern (spherical rhombic dodecahedron) on the surface of a sphere.

1. Place a mark anywhere on a sphere, call it your North Pole.

2. Flip it 180 so that your mark is directly down and place another mark at what is now the very top, and call this mark your South.

3.Hold your poles level to the ground, for bearings and place a third mark directly between.

4. Place a 4th mark opposite your third.

5.Place a 5th mark directly centered between your current 4 and a 6th mark directly opposite this, so you wind up with a Norh Pole mark, a South Pole mark, and 4 equally spaced marks around the equator.

6. Now you need eight more marks to fully describe the vertices of the diamonds. You have just placed the 6 four-point vertices and need to now place the 8, 3 point vertices.

7. Take the sphere and put North directly up, you will see the North Pole and any two of the equatorial marks make a triangle. Place a mark in the center of this triangle, equal distance from the North Pole, the one equatorial point and the other.

8. Repeat this procedure three more times related to the North Pole and the equatorial points as you turn the Sphere 90 degrees each time.

9. Flip so the South pole is up, and place your next four in the same manner as in steps 7 and 8.

10. Now draw four lines down from the North pole, one to each of the marks you made in steps 7 and 8.

11. Flip and draw four lines down from the South pole to the marks you made in step 9.

12. Pick an equatorial mark, which is a four point and visualize an x on this mark extending to the two nearest ends of the lines coming down from the North, and the two nearest ends of the lines coming up from the south. Draw this x.

13 Turn 90 degrees along the equator to the next mark and draw the x again.

14. Turn 90 degress along the equator and draw the x again.

15. Turn 90 degress and draw the x again.

Second procedure.

1. Create an object, (like a string or toothpick) that is one diameter of your sphere, long, with a reference mark at its middle.

2. Make an X on your sphere, marking the length of your object and the middle and then an other line crossing the first at 90 degrees. Calll this your North 4 point, with attending lines.

3. Make a South Pole X in the same manner, so that the lines are headed for, but of course will not reach, the North Pole lines (Visualize two great circles bisecting each other)

4. You will find your diameter marking object exactly reaches between the tip of a line extending from the North Pole, and the next line 90 degrees over, coming up from the South. Draw this line, and make an X out of it, by placing your marker at its center turned 90 degrees. (You will see this conviently, also lines up with the ends of a line coming down from the North and one up from the South.)

5. Make the other three Xs around the equator.

In both cases you should wind up with the 12 diamonds describing the Spherical Rhombic Dodecahedron.

There are other ways of course to get to the figure, as we have described, finding the center points of the diamonds with the vertices of the cubic octahedron and so forth, but these two methods are the quickest and easiest I have found so far.

Regards, TAR

Edited by tar

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Imatfaal,

I thought it interesting today to think of the various shapes and trials and thinking that led me to the pingpong ball close packing and the cubic octohedron, and the figure I learned through this thread was the Spherical Rhombic Dodecahedron, and the Janus and the 12 sections of the sphere, and the cube and the octahedron, and finally the tetrahedron sphere and the six pennies around a central penny that I started with. Interesting, because I realized I had made the tetrahedron sphere when I was 10 years old, taking the the ball of gum out of my mouth and pressing it between the thumb and forefinger of both hands.

And here your picture is of the interlocking tetrahedron.

Long way to go, to get to the figure so easily and naturally formed by a 10 year old.

Regards, TAR

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While cutting the diamonds off the surface of a sphere I accidently came up with a 1 faced, 1 edged, 2 verticed figure that I started a thread on, and continued to investigate.

I can make it at will now (based on a tetrahedron,) and it has a definite relationship to the twelve sections of the sphere. There are left handed and right handed versions of the thing, and you can "start" the thing on any of the twelve sections, in each of the four orientations possible.

The numbering scheme I suggested earlier can be readily applied to this figure, and the imprint of this figure on a 2D surface.

The Red hexagonal plane describes a nice semicircular arc

(open to the NW) (1,6,5,4,3,2)

The Blue hexagonal plane describes a nice semicircular arc

(open to the SE) (11,3,8,12,6,10)

The Yellow hexagonal plane describes an S to the left and down

(1,7,8,4,9,10)

And the Green hexagonal plane describes a S tilted to the right and up

(11,9,5,12,7,2)

Regards, TAR

(sorry my NW and SE, up and to the right is all 90 degrees off, I took the picture after rotating the book 90 degrees CW)

Edited by tar

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Was cutting a Sedge into its 12 sections. Just got the first section cut out and realized they were not going to all be identical. Then I thought that was because some edges of a Sedge section were cut from the edge and some from the face. Then I realized I could not have that issue if I never rounded the three edges required to make the Sedge in the first place...

Don't remember if I ever made the 12 sections of the tetrahedron. If not, I will soon and post them.

Seems those twelve sections should be very interesting with interesting properties. Might even be a way to pack space with them.

Just a waking thought. Will pursue it later.

Regards, TAR

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Cut up a 1/4 pound clay tetrahedron into the 12 sections and replaced one Sedge section with one Tetrahedron section and vise-a-versa.

Show the twelve sections.

Then the surprise. I put the sections outside face to outside face to see how they would pack, and got six peices that unexpectedly fit together as shown. 10 of the pieces completed a circular pattern with a flat side. The other two pieces made the 6th piece which fit nicely upside down in any one of the five "places" for it. Seems the sections will pack nicely.

Regards, TAR

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somehow the sections wind up with that 36 degree angle. Two make 72 degrees, 5 make 180 and 10 make 360.

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tar :

12 identical sections of the Sphere as you have indicated with 12 Squares drawn on the Spherical surface and if cut off at the 14 Points of intersections [6 intersections having 4 lines meeting and 8 intersections having 3 lines meeting] results in 12 rhombus shapes of the Polyhedron left.

This is precisely the Polyhedron which will be created by BOXING THE CUBE as I mentioned elsewhere. That is , planes are attached so as to create a 4 lined ROOFS on each of the 6 Faces of the Cube in a SYMMETRICAL WAY.

But alas it will have to retain the eight 3 lined Corners of the parent Cube though with flatter non- 90 Deg Roof Angles lines rotated from the Original Cube.

Because of this eight intersections have 3 lines meeting points [just like the cube's corners. Along with these 8 six more newly erected 4 lined corners one on top of each face of the Cube adds up to 14 Corners / Intersections.

Because of these 3 line / 4 line corners [each rhombus having 2 four lined corners and 2 three lined corners] we have Rhombus shapes and not Squares on the Polyhedron. I need to confirm what should be their Angles.

Therefore, Boxing a Tetrahedron Symmetrically produces a perfect Cube Boxing a Cube Symmetrically produces a 12 faced Polyhedron with equal Rhombus faces.

If we bloat up each Polyhedron to Spherical Surfaces we have 4 Triangles drawn on the Sphere for tetrahedron, 6 Squares drawn for Cube and 12 Squares drawn for this 12 faced Rhombus hedron.

Splitting each square face of the Cubic bloat-up will produce 24 faced Polyhedron.

Strangely, there is no way of producing 8 equal Square faces on the Sphere, Using 2 tetrahedrons in a mirror tilt may produce 8 equidistant points on the surface of the Sphere but connecting them may again result in Triangular shapes and not squares.

Regards

Edited by Commander

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Commander,

Post 70 in this thread shows my version of your roofed cube. The divisions are the same in terms of internal angles (90 at the 4 points and 120 at the three points.) Except to retain the 12 divisions you ignore the edges of the cube, and count half a diamond as your 1/4 roof on one side, and combine that with its neighboring 1/4 roof on the touching side.

I will have to accept your statement that you can not make 8 equal diamonds. Seems the 12 sections work out so nicely to divide the tetrahedron, the octahedron, the cube and the sphere, it would be unlikely to suspect that another number, other than 12 would work as well.

I had not been able to see the relationship of the cube octahedron I started with, to the dodecahedron with the pentagonal sides. Perhaps now that using 10 out of the twelve tetraheral sections to make that five sided figure a few posts ago, there might be a relationship, after all. However, the spherical rhombic dodecahedron, is still the better of two dodecahedrons, as far as I am concerned.

It has the cube, octahedron and tetrahedron built in.

It has the 4 intersecting hexagonal planes.

It has the 3 intersecting square planes.

The diamonds can be divided in a grid pattern, seemingly proportionally around the hexagonal planes, into degrees, minutes, seconds and fractions of seconds, which you can not do with the pentagonal dodecahedron.

If you imagine a ball at the center of the figure, 12 balls of the same diameter will fit exactly around the center one, if placed exactly in the middle of each of the 12 diamonds. This pattern of hexagonal and square planes, that is thusly indicated, can be extended outward indefinitely.

Each of the sides of the diamonds appears to be exactly one radius of the sphere that the spherical rhombic dodecahedron is drawn onto.

Any one of the said 12 sections of the sphere can replace any one of the 12 sections of the cube, the octahedron and the tetrahedron, in terms of the sections' internal angles. Because of this, mapping the surface of any of the shapes to the other could be done, in terms of the designations of position on the sphere according to the hexagonal plane 360 degree grid model shown earlier.

Regards, TAR

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Commander,

"Strangely, there is no way of producing 8 equal Square faces on the Sphere, Using 2 tetrahedrons in a mirror tilt may produce 8 equidistant points on the surface of the Sphere but connecting them may again result in Triangular shapes and not squares."

I took this as a challenge and was thinking this morning about it. I think I succeeded in making such an "Impossible" division of the sphere into 8 four sided figures, with all four sides being the same length, and on the surface of the sphere, and I believe each line is on a great circle.

I will make it again and take pictures of it, tonight.

TAR