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Post by Ron Walker on Oct 21, 2022 9:13:47 GMT -7
Posted by: Scott T Mar 7 2021, 05:42 AM 1. Designing an optical star ball projector based on a truncated icosahedron.
This thread is a natural progression from Experimental Projectors (three different ones). The aim is to design and build a reasonably accurate and elegant home-made optical projector based on a truncated icosahedron design and to do it step by step with no unexplained ‘magic jumps’ so that it can be reproduced by other amateurs (or more likely, adapted and improved upon).
Sir Gare’s mantra has been applied i.e. many hours of thought precede 1 hour of action.
I hope this will be either interesting or even useful to someone out there and I apologise in advance for the bits where it goes over well-trodden ground. The physical construction will probably take a while and posts will be a bit sporadic with long gaps as they always are with me. Maybe it will be ready for the centenary in 2023!
There may more detail than strictly necessary in places. If there is not enough detail or if you spot mistakes let me know.
I would love to be able to dabble in the fiendish mathematical wizardry that Paul Bourke and Jamie Hicks are using on the ‘star plate’ thread but I think it is out of my league. I believe I have found a path to a similar end result using more basic techniques –i.e. at the level of multiplying, dividing & plugging in numbers to some basic formulae carefully lifted from the four corners of the internet. I do make extensive use of a spreadsheet - I happen to use Excel but any equivalent will do.
The individual steps should be understandable and relatively straight forward but a complex, wonderful object should eventually emerge at the end.
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Post by Ron Walker on Oct 21, 2022 9:19:50 GMT -7
Posted by: Scott T Mar 7 2021, 05:46 AM 2. Some notes about truncated icosahedrons. The projector will be based on a truncated icosahedron – the underlying approach used in the original Zeiss projector star balls and copied by others ever since. The basic design is skimmed over briefly in a few books and articles. I happen to have Captured Stars by Heinz Letsch and The Planetarium and atmospherium by O. Richard Norton. These are great books but I guess the authors were not really intending to write manuals for oddball amateurs to try to reproduce what Zeiss achieved. Consequently, there are a several important or helpful details missing from their descriptions– forgive me if these bits are more obvious to you than they were to me but they are important for being able to really understand what is going on. One important note: I think the Zeiss star plates were based on a form called an isodistant truncated icosahedron not the common or garden truncated icosahedron. So what is the difference and is it a big deal? The standard truncated icosahedron is the sort that pops up on google if you search for truncated icosahedron and looks like a normal football/ soccer ball. To ‘make’ one you start with an icosahedron 'a' in Figure 1. Apparently, it is known as a Platonic solid and is made of equilateral triangles. If you cut off (‘truncate’) small pyramidical caps at the vertices (corners) 'b' you are left with a truncated icosahedron 'c' or a basic football shape (soccer ball if you are on the other side of the pond!) This ‘standard’ truncated icosahedron is a form composed of 20 hexagons and 12 pentagons. It has 60 vertices (corners). The pentagons are about half the size of the hexagons and all their edges are exactly the same length.
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Post by Ron Walker on Oct 21, 2022 9:26:30 GMT -7
Posted by: Scott T Mar 7 2021, 05:54 AM BUT look closely – this is not the same shape as the pictures of what Zeiss did according to Norton or Letsch’s book (Figure 2). The hexagons are roughly the same size as the pentagons and the hexagons are definitely lopsided. What is going on? The lopsided hexagons are really obvious in Letch’s picture of the division of the sky (Figure 3).
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Post by Ron Walker on Oct 21, 2022 9:28:02 GMT -7
I previously did not pay much attention to why: The best ‘accessible’ explanation I can find comes from a paper by I. Pieter Huybers (2006) which is actually about footballs: In search of the roundest soccer ball, International Conference On Adaptable Building Structures Eindohven. It is also where I have copied some of the pictures from. Dr Huybers paper contains a far more rigorous explanation than the following attempt and is worth a read, but in summary- When you are cutting the pyramid like shapes off of your icosahedron, you are free to decide how big those pyramid-like offcuts are. The standard truncated icosahedron is made when the base of the pyramid is cut one third of the way down the triangular face of the icosahedron Then, all the edges of the hexagons are exactly the same length. If you decide to cut off slightly bigger pyramid like shapes by making your cuts a bit further down the edge of the triangle, your resultant pentagons will get bigger and your hexagons will become smaller and more lopsided (figure 4 and 5). Apparently, these lopsided hexagons are called ditrigons.
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Post by Ron Walker on Oct 21, 2022 9:31:45 GMT -7
Posted by: Scott T Mar 7 2021, 06:10 AM That seems odd - why would you want to make lopsided hexagons? If you cut off the pyramids in exactly the right place, the surface area of the lopsided pentagons and hexagons can be made equal – so Zeiss chose to divide up the sky into 32 portions of equal area rather than 20 big portions and 12 smaller portions! That makes perfect sense. That is why Zeiss chose an isodistant truncated isocahedron. Further trivia: There may have been a second more subtle reason for choosing the isodistant version. In a standard truncated icosahedron all the corners (vertices) of the pentagons and hexagons touch a perfect imaginary sphere that perfectly encloses them. However, because the pentagons and hexagons are flat planes and the hexagons are much wider than the pentagons, their middles (centroids) lie at slightly different depths just inside the imaginary sphere (see cross sectional sketch below). Or put another way, the centroids of each face are at slightly different distances from the centre/origin of the icosahedron. In contrast, as the name suggests, in the isodistant version, the pentagons and hexagons are basically the same size and centres of both the pentagons and hexagons can be thought of as sitting on a single imaginary sphere i.e. the centroids of both the hexagons and the pentagons are at exactly the same distance from the centre of the sphere (isodistant literally means 'the same distance'). That is practically quite helpful when the middle of your hexagons and pentagons support identical projection lenses in a star ball projector. Incidentally, in a real football, air pressure stretches out the skin material so the centres of the pentagons and hexagons push out and the whole thing looks spherical to the naked eye and few people worry overly about it. If you make an isodistant form of football it means that your ball is actually slightly (a few %) more spherical than the standard form. It looks like it took football manufacturers roughly 80 years to catch up with Zeiss and the ‘best’ balls are now isodistant designs. So that begs a question – should we use a standard or an isodistant icosahedron? I am going to leave that question hanging for a while and tackle a different issue.
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Post by Ron Walker on Oct 21, 2022 9:33:20 GMT -7
Posted by: Scott T Mar 7 2021, 06:17 AM I am also going to pause and celebrate graduating from interstellar medium to dust disk with my 100th post!
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Post by Ron Walker on Oct 21, 2022 9:34:12 GMT -7
Posted by: mrgare5050 Mar 7 2021, 07:31 AM
QUOTE(Scott T @ Mar 7 2021, 01:17 PM) *I am also going to pause and celebrate graduating from interstellar medium to dust disk with my 100th post!
write that book!
I was once a dust disk - it never got any better lol. those were the days
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Post by Ron Walker on Oct 21, 2022 9:34:59 GMT -7
Posted by: Ron Walker Mar 7 2021, 12:54 PM So good to have a good new discussion thread for builders of planetariums. I, for one, will follow this with great interest.
I'm wondering if the drawings in the book are accurate or just slightly off to make the two dimensional printing look more 3D.
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Post by Ron Walker on Oct 21, 2022 9:35:18 GMT -7
Posted by: Ron Walker Mar 7 2021, 12:59 PM After careful study of the above, I vote for a isodistant icosahedron.
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Post by Ron Walker on Oct 21, 2022 9:36:36 GMT -7
Posted by: Scott T Mar 7 2021, 01:23 PM Thanks Ron and Sir Gare - on with the story. 3. The tessellation problem. I struggled with the following conundrum for a while. A projection lens will project an upside down image. Anyone who uses old fashioned 35mm side projectors knows to put the slides in upside down and the problem is solved. Similarly, inverted projection is not a problem with hexagonal sections – just invert the star plates and the projected image will be fine. However, when the pentagons are inverted surely they would lead to horrible gaps in the sky (as shown in the diagram).
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Post by Ron Walker on Oct 21, 2022 9:41:16 GMT -7
Posted by: Scott T Mar 7 2021, 01:26 PM I thought perhaps the pentagonal sections had an extra inverting lens or used some other method for overcoming the problem. As ever, Ron pointed straight toward the answer on this forum - that each star plate actually projects a circular patch of sky which must overlap slightly. Using overlapping circular areas reduces the tessellation problem to simply eliminating any ‘duplicated’ stars in the overlapping sections. In my defence, the text in Letsch was not particularly clear: "The 12 corners of an icosahedron being truncated, a body will be formed, the surface of which consists of 20 hexagons and 12 pentagons. Round these 32 single surfaces circles of approximate diameters may be described, if the intersectional planes are in proper positions. The solid edges projected on to the projection dome show the division of the sky of fixed stars." Maybe something was lost in the translation from German to English! Norton states that the star fields exactly fits the adjacent one - so I still have a seed of doubt about the exact approach Zeiss took. To summarise, the star plates appear to be equally sized overlapping circles centred on the pentagons and hexagons of the isodistant truncated icosahedron. Thanks go to Lengyel and Tarnai in a remote corner of the internet for supplying two photos that paint a thousand words:
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Post by Ron Walker on Oct 21, 2022 9:43:00 GMT -7
Posted by: Scott T Mar 8 2021, 12:50 PM 4 A short diversion: The disappearing eccentric problem Another thing that Zeiss did for their dumbbell style projectors was to correct the ‘eccentric problem’. If the starball is positioned in the exact centre of the dome it is easy to imagine that in order to project one star at the zenith the dome and one at the equator you simply need two holes – one pointing straight up and one pointing at the equator with an angle of 90 degrees between them.
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Post by Ron Walker on Oct 21, 2022 9:44:35 GMT -7
Posted by: Scott T Mar 8 2021, 12:53 PM Figure 39 from Letsch’s book illustrates the ‘eccentric problem’ for dumbbell projectors. If the starball is off centre (eccentric) with respect to the dome then you need to increase the angle slightly between those two stars if they are to continue to point at the pole and the equator. It appears Zeiss carried out this angular correction by photographing the relevant portion of a star map from an appropriate angle and then using the photograph as a basis for plotting out the star plate. I hazard a guess that modern approach would be to use clever maths and I strongly suspect it would be too complicated for me. As the name of this section suggests – the ‘eccentric problem’ disappears if your design has the star ball positioned in the exact centre of the dome – so that is one easy decision to make at this stage: the star ball will be designed to be in the centre of the dome! Problem solved. With one eye on the future and the nagging doubt that the urge to make a dumbbell system may one day become stronger and stronger (this hobby is a disease) it would be interesting to know how big a problem this really is but that is an issue for another day.
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Post by Ron Walker on Oct 21, 2022 9:45:25 GMT -7
Posted by: moonmagic Mar 8 2021, 07:49 PM
QUOTE(Scott T @ Mar 8 2021, 02:53 PM) *Figure 39 from Letsch’s book illustrates the ‘eccentric problem’ for dumbbell projectors. If the starball is off centre (eccentric) with respect to the dome then you need to increase the angle slightly between those two stars if they are to continue to point at the pole and the equator.
It appears Zeiss carried out this angular correction by photographing the relevant portion of a star map from an appropriate angle and then using the photograph as a basis for plotting out the star plate. I hazard a guess that modern approach would be to use clever maths and I strongly suspect it would be too complicated for me.
As the name of this section suggests – the ‘eccentric problem’ disappears if your design has the star ball positioned in the exact centre of the dome – so that is one easy decision to make at this stage: the star ball will be designed to be in the centre of the dome! Problem solved.
With one eye on the future and the nagging doubt that the urge to make a dumbbell system may one day become stronger and stronger (this hobby is a disease) it would be interesting to know how big a problem this really is but that is an issue for another day.
"MOON" says WOW!
I read all your posts and then needed to re-read some of it to let parts sink in. I learned a whole lot. Explanations and research are WELL DONE! KUDOS.
During the entire time I was reading these great posts I was also recalling the story of the design of the early Spitz projectors. If I recall correctly Both Spitz and Einstein were neighbors when they lived in NJ. They were kitchen table friends who drank coffee, talked, and pondered the world. Spitz gave Einstein credit when he helped him solve his early issue of not being able to procure a metal thin enough (to not cause a tunnel effect for each drilled star) yet strong enough to hold a round shape that he could drill star holes into. Seems Einstein mentioned that the closest shape to being round was that of a dodecahedron. (12 sided pentagons, 20 vertices, 30 edges) (Keeping in mind Spitz was trying to use ONE single star globe, not a dumbbell design) Can one use larger lenses along with larger star plates in such a configuration or would this produce the problem of spaces in between some star plates (where their edges join)?
As an aside: The Government (ours) uses this same soccer ball design to cover all the Doppler radar antennas. (Web indicates the parabolic dish is about 30' in diameter and turns in both azimuth and elevation).
Also on another matter, depending on the light source you intend to employ; some extra room inside makes it easier to replace said source and allows more air to dissipate the heat without need of an internal fan UNLESS you go with small LED's. mm Posted by: moonmagic Mar 8 2021, 07:49 PM
QUOTE(Scott T @ Mar 8 2021, 02:53 PM) *Figure 39 from Letsch’s book illustrates the ‘eccentric problem’ for dumbbell projectors. If the starball is off centre (eccentric) with respect to the dome then you need to increase the angle slightly between those two stars if they are to continue to point at the pole and the equator.
It appears Zeiss carried out this angular correction by photographing the relevant portion of a star map from an appropriate angle and then using the photograph as a basis for plotting out the star plate. I hazard a guess that modern approach would be to use clever maths and I strongly suspect it would be too complicated for me.
As the name of this section suggests – the ‘eccentric problem’ disappears if your design has the star ball positioned in the exact centre of the dome – so that is one easy decision to make at this stage: the star ball will be designed to be in the centre of the dome! Problem solved.
With one eye on the future and the nagging doubt that the urge to make a dumbbell system may one day become stronger and stronger (this hobby is a disease) it would be interesting to know how big a problem this really is but that is an issue for another day.
"MOON" says WOW!
I read all your posts and then needed to re-read some of it to let parts sink in. I learned a whole lot. Explanations and research are WELL DONE! KUDOS.
During the entire time I was reading these great posts I was also recalling the story of the design of the early Spitz projectors. If I recall correctly Both Spitz and Einstein were neighbors when they lived in NJ. They were kitchen table friends who drank coffee, talked, and pondered the world. Spitz gave Einstein credit when he helped him solve his early issue of not being able to procure a metal thin enough (to not cause a tunnel effect for each drilled star) yet strong enough to hold a round shape that he could drill star holes into. Seems Einstein mentioned that the closest shape to being round was that of a dodecahedron. (12 sided pentagons, 20 vertices, 30 edges) (Keeping in mind Spitz was trying to use ONE single star globe, not a dumbbell design) Can one use larger lenses along with larger star plates in such a configuration or would this produce the problem of spaces in between some star plates (where their edges join)?
As an aside: The Government (ours) uses this same soccer ball design to cover all the Doppler radar antennas. (Web indicates the parabolic dish is about 30' in diameter and turns in both azimuth and elevation).
Also on another matter, depending on the light source you intend to employ; some extra room inside makes it easier to replace said source and allows more air to dissipate the heat without need of an internal fan UNLESS you go with small LED's. mm Posted by: moonmagic Mar 8 2021, 07:49 PM
QUOTE(Scott T @ Mar 8 2021, 02:53 PM) *Figure 39 from Letsch’s book illustrates the ‘eccentric problem’ for dumbbell projectors. If the starball is off centre (eccentric) with respect to the dome then you need to increase the angle slightly between those two stars if they are to continue to point at the pole and the equator.
It appears Zeiss carried out this angular correction by photographing the relevant portion of a star map from an appropriate angle and then using the photograph as a basis for plotting out the star plate. I hazard a guess that modern approach would be to use clever maths and I strongly suspect it would be too complicated for me.
As the name of this section suggests – the ‘eccentric problem’ disappears if your design has the star ball positioned in the exact centre of the dome – so that is one easy decision to make at this stage: the star ball will be designed to be in the centre of the dome! Problem solved.
With one eye on the future and the nagging doubt that the urge to make a dumbbell system may one day become stronger and stronger (this hobby is a disease) it would be interesting to know how big a problem this really is but that is an issue for another day.
"MOON" says WOW!
I read all your posts and then needed to re-read some of it to let parts sink in. I learned a whole lot. Explanations and research are WELL DONE! KUDOS.
During the entire time I was reading these great posts I was also recalling the story of the design of the early Spitz projectors. If I recall correctly Both Spitz and Einstein were neighbors when they lived in NJ. They were kitchen table friends who drank coffee, talked, and pondered the world. Spitz gave Einstein credit when he helped him solve his early issue of not being able to procure a metal thin enough (to not cause a tunnel effect for each drilled star) yet strong enough to hold a round shape that he could drill star holes into. Seems Einstein mentioned that the closest shape to being round was that of a dodecahedron. (12 sided pentagons, 20 vertices, 30 edges) (Keeping in mind Spitz was trying to use ONE single star globe, not a dumbbell design) Can one use larger lenses along with larger star plates in such a configuration or would this produce the problem of spaces in between some star plates (where their edges join)?
As an aside: The Government (ours) uses this same soccer ball design to cover all the Doppler radar antennas. (Web indicates the parabolic dish is about 30' in diameter and turns in both azimuth and elevation).
Also on another matter, depending on the light source you intend to employ; some extra room inside makes it easier to replace said source and allows more air to dissipate the heat without need of an internal fan UNLESS you go with small LED's. mm Posted by: moonmagic Mar 8 2021, 07:49 PM
QUOTE(Scott T @ Mar 8 2021, 02:53 PM) *Figure 39 from Letsch’s book illustrates the ‘eccentric problem’ for dumbbell projectors. If the starball is off centre (eccentric) with respect to the dome then you need to increase the angle slightly between those two stars if they are to continue to point at the pole and the equator.
It appears Zeiss carried out this angular correction by photographing the relevant portion of a star map from an appropriate angle and then using the photograph as a basis for plotting out the star plate. I hazard a guess that modern approach would be to use clever maths and I strongly suspect it would be too complicated for me.
As the name of this section suggests – the ‘eccentric problem’ disappears if your design has the star ball positioned in the exact centre of the dome – so that is one easy decision to make at this stage: the star ball will be designed to be in the centre of the dome! Problem solved.
With one eye on the future and the nagging doubt that the urge to make a dumbbell system may one day become stronger and stronger (this hobby is a disease) it would be interesting to know how big a problem this really is but that is an issue for another day.
"MOON" says WOW!
I read all your posts and then needed to re-read some of it to let parts sink in. I learned a whole lot. Explanations and research are WELL DONE! KUDOS.
During the entire time I was reading these great posts I was also recalling the story of the design of the early Spitz projectors. If I recall correctly Both Spitz and Einstein were neighbors when they lived in NJ. They were kitchen table friends who drank coffee, talked, and pondered the world. Spitz gave Einstein credit when he helped him solve his early issue of not being able to procure a metal thin enough (to not cause a tunnel effect for each drilled star) yet strong enough to hold a round shape that he could drill star holes into. Seems Einstein mentioned that the closest shape to being round was that of a dodecahedron. (12 sided pentagons, 20 vertices, 30 edges) (Keeping in mind Spitz was trying to use ONE single star globe, not a dumbbell design) Can one use larger lenses along with larger star plates in such a configuration or would this produce the problem of spaces in between some star plates (where their edges join)?
As an aside: The Government (ours) uses this same soccer ball design to cover all the Doppler radar antennas. (Web indicates the parabolic dish is about 30' in diameter and turns in both azimuth and elevation).
Also on another matter, depending on the light source you intend to employ; some extra room inside makes it easier to replace said source and allows more air to dissipate the heat without need of an internal fan UNLESS you go with small LED's. mm
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Post by Ron Walker on Oct 21, 2022 9:55:06 GMT -7
Posted by: Scott T Mar 9 2021, 09:52 AM Many thanks Moon!
It would be great to eavesdrop at that kitchen table.
In answer to your question: Can one use larger lenses along with larger star plates in such a configuration or would this produce the problem of spaces in between some star plates (where their edges join)?
I think you can- you don't necessarily need a 32 plate arrangement. There is a delicate interplay between plate size, diameter and focal length of the lens and distance to the dome etc and somewhere there is an optimum combination. Ultimately, it would be hard to focus an extremely large flat plate on a curved surface without fancy optics. I have a future section planned for those issues but it is not quite written up yet.
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