Monday, September 7, 2009
Monday, July 28, 2008
Columbia Tragedy
The following discussion was between myself (Bobhexa) and other members and guests of Eng-tips.krd (Civil)
4 Mar 03 16:12
Are there experts in this forum who might care to comment on the servicability of the shuttle heat shield design? Since the future of the shuttles will become a subject of public debate, and maybe even a vote in Congress, those of us who are not as well informed should try to get educated. It appears, from reading risk assessments available on NASA web pages, that the current shuttle heat shield is a major weak link in the system. Changes in operating procedures and provisions for repairs may allow existing shuttles to fly some more but ultimately, it seems to me, they should be phased out in favour of space craft with more robust (and forgiving?) skin technology. Would you agree that the risk of shuttle skin failure is sufficient enough to make a design change the appropriate action?
bobhexa (Structural)
10 Mar 03 1:53
I would agree that the risk of shuttle skin failure is sufficient enough to make a design change the appropriate action? One wonders if one could not dispense altogether with the ablative tiles and introduce a new skin design wherein the core is ventable and coolants (liquid hydrogen?) are pumped through the core during re-entry to serve as a heat sink.The weight saving of the tiles should allow for the weight of the coolant. The nozzles of the rocket engines are regeneratively cooled so why not extend this concept to the nose cone.
Metalguy (Materials)
10 Mar 03 4:21
Good idea. Makes me wonder if an internal water spray system might work. It would spray water on the internal skin surfaces of the critical areas, but only if detectors sensed above-normal temps. Shouldn't be too hard to do some tests to find out how much water would be required to save the present alum. skin and a possible future Ti one. Water has a very high latent heat of evap, so a "moderate" amount sprayed in the right places might be able to "save the day".
bobhexa (Structural)
10 Mar 03 5:14
The water spray system is a cool idea.I worry about how one might contain the steam that would be generated. Does the shuttle carry any reserve amount of liquid hydrogen upon its re-entry or is it jettisoned? Seems that it could be circulated through a metal foam sandwich panel that is brazed together rather than glued. Maybe three plies. Are there titanium open metal foams available with say a 40 pores per inch in order to maximise the brazing area of core to the titanium skins to make a nose cone sandwich panel a true monocoque assembly that is fully ventable. Ideas on coolants would be welcomed as I personally don't trust the use of cryogenics after the cryo-pumping episodes of the x33. Perhaps the water may do the job as well as any other coolant. Superheated espresso coffee will be served upon landing.
IRstuff (Aerospace)
10 Mar 03 15:02
Have you done the analysis already? There's a heat load of 20 kW/m^2 for probably over 600 sec. On a 50 ft square area, that's about 3 billion joules.
TTFN
bobhexa (Structural)
10 Mar 03 23:37
How many gallons of freon 16 does it take to quench 3 billion joules? Serious answers only please!
Metalguy (Materials)
11 Mar 03 1:07
To make this more realistic, I think we have to assume a small area of skin exposure-say 0.1 sq. meter. So at worse all we have to do is absorb 2,000 watts for 10 min.-should be easy if the spray system is designed with some thought.Since we could tolerate quite a bit of temp. rise, we might only have to absorb half of that.
bobhexa (Structural)
11 Mar 03 5:36
Running the coolant through the core would allow the skin to which the ablative tiles are affixed to be directly cooled. That's where we need the chill factor. The core would need to have major ventability and with the metal foam cell size at 10ppi I wonder whether that would allow a sufficient flow rate. The increase in cell size unfortunately diminishes the strength of the skin to core bond. If lateral vent channels were discretely cut into the titanium metal foam core a higher density foam could be used. Back to high strength.!! The original thought was that this was to be a replacement for the use of ablative tiles but if it couldn't handle that job it would still be a fine back-up or an in-tandem system. It may also be a stand alone system. With a three layered system with each layer having say 3/8th" square cross sectioned ventways cut into a 1/2' thick titanium foam metal core at a spacing of say 1" would that allow a sufficient flow rate of coolant through the panel. That depends on the coolant I guess. I wonder how freon would do?
Sunday, July 20, 2008
Spacecraft Engineering discussion
The following discussion was between myself (Bobhexa) and other members and guests of Eng-tips.
djaytch99 (Structural)
28 Dec 01 18:38
My question concerns the X-33 space plane and the use of honeycomb cores in composites. My understanding of the situation is that the program was cancelled due mainly to the failure of the composite fuel tanks, which, after core delamination was detected, it was decided to use a different material thus increasing the weight, resulting in reassessment and eventual cancellation of the program. I'm sure that is an over-simplification of the circumstances, but what puzzles me is that for many years now state-of-the-art in honeycomb technology has always revolved around the industry standard: i.e strips of metal or similar, glued together to form hexagonal open-ended tubes of varying diameter and length. My interest in this subject stems from the fact that due to my own discovery of a new method of producing honeycomb, I have been searching the patent databases to see what the competition has to offer. Imagine my surprise to find that there are literally hundreds of ways to form honeycomb, many of which would certainly do a better job than current designs. It doesn't seem to matter whether it's aerospace, powerboats, automobiles or whatever, good ole' state-of-the-art honeycomb is where it's at, and always has been! So, what is the problem? David
bobhexa(Structural)
10 Jan 02 5:02
Hi David,The problem encountered by the X33 was due mainly to cryo-pumping which could really only occur in a honeycomb, closed cell, core material...which of course is the current state of the art. I sense a touch of cynicism?I personally feel that if engineers worth their salt could realise that the continued unabashed use of standard honeycomb in aerospace, powerboats, automobiles or high speed trains is an archaic pavlovian response ...... "Well, we've always used ribbons in our honeycomb". Good engineering now demands us to re-evaluate and realise the major disadvantages of honeycomb which have been adopted and accepted all in the name of saving weight.Catastrophic failure without regular maintenance checks is guaranteed.Incidentally all these applications generally involve people and fast moving objects. Someone was asking earlier on, in the aeronautical forum, as to whether aircraft are really glued together. Well in the main that is true.honeycomb is probably 25% glue. That is the nub of the problem.If only the glue would stay stuck there really wouldn't be a crisis.The closed cell honeycomb composite panels in craft which undergo ambient pressure changes combined with the minimal surface bonding area of laminates to core is tempting fate. So if the glue had a larger bonding area then all would be fine and delamination would not be the focus.If somebody could invent a structural panel that was ventable, formable, as light or lighter than present art and that had a mere 500 percent increase in bonding area we will get to mars and back and the Americas cup yacht race will be won by a craft with a foresail almost up front without causing the boat to snap in half.The present honeycomb model was discovered by an undergraduate doodling with strips of cardboard and he started the ball rolling.........it is now time for engineers to start thinking more laterally and move beyond an archaic solution,Best regards Bob (:]
vonlueke (Structural)
11 Jan 02 0:04
How about aluminum foam core? Would that work? Thanks.
bobhexa (Structural)
13 Jan 02 3:03
Aluminum foam core is an extremely good solution in that it has the rough texture plus a far larger surface area which promotes a good bonding of the face sheets.In terms of lightness of material it wins hands down, in so far that it has no glue required to add to the weight. Ventable is a moot point depending on which process is used.In one method a foaming agent is mixed with powdered aluminum and produces ,when heated, gases which form bubbles within the molten aluminum. It solidifies into a block of aerated aluminum which does not vent. A bit like Aero Chocolate! Another method involves filling a mould with something like rock salt (engineers worth their salt?)or clay balls (no comment) compressing it all so the fill will snuggle up tightly to its neighbour and then injecting molten aluminum into the mold and ,when cool, one then washes out the salt/clay and .... "bingo" ........ a lightweight aluminum sponge which vents. One can imagine the number of steps required to make this a very expensive method of manufacture...... never the less it could be done . Let us now focus on formability...by all accounts metal foam core using the 2nd method. which allows venting, would need to be milled from a solid block to produce a panel with surface curvature as it does not naturally flex. So the results are in:Ventable.......................... Yes. Formable.......... ............... No. Lightweight....................... Yes. 500% increase in bonding area...... Yes.
wktaylor (Aeronautics)
17 Jan 02 18:05
The X-33/-34 suffered from several major problems that finally came together, causing melt-down.The Cryonic tank for liquid [slush?] hydrogen was mandatory for weight savings [bigger volume requirement - relative to the LOX tank].The thermal and pressure stresses destroyed joints between major assemblies. Unfortunately, these joints would have been uninspectable [within reason] "in service".Many areas of the program had major cost over-runs, but the "problems were getting solved". HOWEVER, the LH2 tank problem wasn't getting solved... and would require massive redesign to correct the deficiencies [engineers painted the design into a very tight corner].Simply put: The design stumbled over too many design obstacles and costs got out-of-hand... and no one stood-up to the political pressure and everyone "avoided" the hard questions... until the obvious tank-design failure sealed it's fate. I believe the McDonnell-Douglass design had great merit... but wasn't "sexy"... or "hi-tech" [old concept]. OH WELL...WKT
bobhexa (Structural)
28 Jun 02 7:24
Eureka!!I have been pondering on the disadvantages of honeycomb in the aerospace industry.......closed cells....minimum bonding areas etc.The main disadvantage of honeycomb type cores is that of delamination. This is caused primarily by the failure of the epoxy adhesives to maintain a bond between the skins and the core because of the very small bonding area that honeycomb cores offer. This is further exacerbated by the fact that honeycomb type cores create pockets of trapped air within the closed cells of the core when the skins are attached. The air pressure experienced at high altitudes is much lower than the trapped air within the cells with the result that the skin is pushed away from the inner core by the air pressure. Ingress of water into already partially delaminated cells at high altitude freeze into ice particles which expand and force the skin to separate from the core. Eventually after many cyclic operations the skin will delaminate. Lightning strikes can cause entrapped moisture to immediately turn to steam with catastrophic results to the integral strength of the panels.I do believe I have the solution to it all.If NASA or the FAA would like to contact me I would be pleased to enlighten them.
djaytch99 (Structural)
28 Dec 01 18:38
My question concerns the X-33 space plane and the use of honeycomb cores in composites. My understanding of the situation is that the program was cancelled due mainly to the failure of the composite fuel tanks, which, after core delamination was detected, it was decided to use a different material thus increasing the weight, resulting in reassessment and eventual cancellation of the program. I'm sure that is an over-simplification of the circumstances, but what puzzles me is that for many years now state-of-the-art in honeycomb technology has always revolved around the industry standard: i.e strips of metal or similar, glued together to form hexagonal open-ended tubes of varying diameter and length. My interest in this subject stems from the fact that due to my own discovery of a new method of producing honeycomb, I have been searching the patent databases to see what the competition has to offer. Imagine my surprise to find that there are literally hundreds of ways to form honeycomb, many of which would certainly do a better job than current designs. It doesn't seem to matter whether it's aerospace, powerboats, automobiles or whatever, good ole' state-of-the-art honeycomb is where it's at, and always has been! So, what is the problem? Davidbobhexa(Structural)
10 Jan 02 5:02
Hi David,The problem encountered by the X33 was due mainly to cryo-pumping which could really only occur in a honeycomb, closed cell, core material...which of course is the current state of the art. I sense a touch of cynicism?I personally feel that if engineers worth their salt could realise that the continued unabashed use of standard honeycomb in aerospace, powerboats, automobiles or high speed trains is an archaic pavlovian response ...... "Well, we've always used ribbons in our honeycomb". Good engineering now demands us to re-evaluate and realise the major disadvantages of honeycomb which have been adopted and accepted all in the name of saving weight.Catastrophic failure without regular maintenance checks is guaranteed.Incidentally all these applications generally involve people and fast moving objects. Someone was asking earlier on, in the aeronautical forum, as to whether aircraft are really glued together. Well in the main that is true.honeycomb is probably 25% glue. That is the nub of the problem.If only the glue would stay stuck there really wouldn't be a crisis.The closed cell honeycomb composite panels in craft which undergo ambient pressure changes combined with the minimal surface bonding area of laminates to core is tempting fate. So if the glue had a larger bonding area then all would be fine and delamination would not be the focus.If somebody could invent a structural panel that was ventable, formable, as light or lighter than present art and that had a mere 500 percent increase in bonding area we will get to mars and back and the Americas cup yacht race will be won by a craft with a foresail almost up front without causing the boat to snap in half.The present honeycomb model was discovered by an undergraduate doodling with strips of cardboard and he started the ball rolling.........it is now time for engineers to start thinking more laterally and move beyond an archaic solution,Best regards Bob (:]
vonlueke (Structural)
11 Jan 02 0:04
How about aluminum foam core? Would that work? Thanks.
bobhexa (Structural)
13 Jan 02 3:03
Aluminum foam core is an extremely good solution in that it has the rough texture plus a far larger surface area which promotes a good bonding of the face sheets.In terms of lightness of material it wins hands down, in so far that it has no glue required to add to the weight. Ventable is a moot point depending on which process is used.In one method a foaming agent is mixed with powdered aluminum and produces ,when heated, gases which form bubbles within the molten aluminum. It solidifies into a block of aerated aluminum which does not vent. A bit like Aero Chocolate! Another method involves filling a mould with something like rock salt (engineers worth their salt?)or clay balls (no comment) compressing it all so the fill will snuggle up tightly to its neighbour and then injecting molten aluminum into the mold and ,when cool, one then washes out the salt/clay and .... "bingo" ........ a lightweight aluminum sponge which vents. One can imagine the number of steps required to make this a very expensive method of manufacture...... never the less it could be done . Let us now focus on formability...by all accounts metal foam core using the 2nd method. which allows venting, would need to be milled from a solid block to produce a panel with surface curvature as it does not naturally flex. So the results are in:Ventable.......................... Yes. Formable.......... ............... No. Lightweight....................... Yes. 500% increase in bonding area...... Yes.
wktaylor (Aeronautics)
17 Jan 02 18:05
The X-33/-34 suffered from several major problems that finally came together, causing melt-down.The Cryonic tank for liquid [slush?] hydrogen was mandatory for weight savings [bigger volume requirement - relative to the LOX tank].The thermal and pressure stresses destroyed joints between major assemblies. Unfortunately, these joints would have been uninspectable [within reason] "in service".Many areas of the program had major cost over-runs, but the "problems were getting solved". HOWEVER, the LH2 tank problem wasn't getting solved... and would require massive redesign to correct the deficiencies [engineers painted the design into a very tight corner].Simply put: The design stumbled over too many design obstacles and costs got out-of-hand... and no one stood-up to the political pressure and everyone "avoided" the hard questions... until the obvious tank-design failure sealed it's fate. I believe the McDonnell-Douglass design had great merit... but wasn't "sexy"... or "hi-tech" [old concept]. OH WELL...WKT
bobhexa (Structural)
28 Jun 02 7:24
Eureka!!I have been pondering on the disadvantages of honeycomb in the aerospace industry.......closed cells....minimum bonding areas etc.The main disadvantage of honeycomb type cores is that of delamination. This is caused primarily by the failure of the epoxy adhesives to maintain a bond between the skins and the core because of the very small bonding area that honeycomb cores offer. This is further exacerbated by the fact that honeycomb type cores create pockets of trapped air within the closed cells of the core when the skins are attached. The air pressure experienced at high altitudes is much lower than the trapped air within the cells with the result that the skin is pushed away from the inner core by the air pressure. Ingress of water into already partially delaminated cells at high altitude freeze into ice particles which expand and force the skin to separate from the core. Eventually after many cyclic operations the skin will delaminate. Lightning strikes can cause entrapped moisture to immediately turn to steam with catastrophic results to the integral strength of the panels.I do believe I have the solution to it all.If NASA or the FAA would like to contact me I would be pleased to enlighten them.
Saturday, July 12, 2008
Pacific Handcrafters Guild Fair July 12-13 2008
We had a truly spectacular day at Thomas Square Park today! Met a whole bunch of new friends, along with good old friends.
Comments were:
"Spectacular Spectacular"
from a gentleman who I tipped handsomely.
"Very Buckminster Fuller-ish"
from a guy who seemed C60-ish;
"Where did you get the copper?"
from Ron, the fine Cop on duty;
"Are you a Geordie?"
from a Glaswegian woman.
The BBC played for their lunch. No starving artists today! :)
Side note:
The BBC is comprised of David, Manny and myself (2 Brits and 1 Chicano).
Our normal spot is at Hank's Cafe, Nuuanu, Chinatown on Tuesday nights where we play Stones, Beatles, Pink Floyd, Elvis, Folk, Blues and beyond. Come down for a hoot!
Wednesday, July 9, 2008
The Hawaiian Green Turtle
This unique hand made sculpture is derived from a geometry between the hexagon and the square to create the shell of the Hawaiian Green Turtle. Serendipitously, the resulting geometry has the perfect openings for the wooden head and limbs to be placed. Only local Hawaiian woods such as koa, mango, milo, mac-nut and lychee are used.The Hawaiian green turtle is an important symbol in the mythology of Hawaiian culture.
The Honu is a Hawaiian symbol of creation, longevity, peace and the spirit within.
In Chinese mythology, the sea turtle also represents wisdom. The shell of the turtle is said to have the markings of heaven and earth upon it. It is also said that the Turtle had risen from the depths with the markings of the eight trigrams on its back. These are used in the I Ching and the Bagwa.
☰☱☲☳☴☵☶☷
The sea turtle mirrors the health of our planet and the struggle for survival for these ancient creatures is in our own hands.
The sea turtle mirrors the health of our planet and the struggle for survival for these ancient creatures is in our own hands.
Tuesday, July 8, 2008
Garnet city. Architectural Dodecahedronism.
Always thought that this would make a beautiful Japanese tea house.
This is a dream of a structure for which I have invented a truss system. I would love to have say 5 acres of land subdivided into hexagonal plots and offer them to the public who would build a rhombic dodecahedron (RD) homestead upon an exact hexagonal grid floorplan. Eventually over time it could be developed into a tetrahedral pyramid as families would add another RD to their dwelling.It may qualify for a humanities grant to research how socio-logically the site would develop.
This is a paperweight model of a Japanese Tea House. Notice the architectural problem that the builder is having trying to put a smaller upper deck onto a larger lower deck. The problem is one of orthogonality. It works so much more elegantly by using rhombic dodecahedrons as seen above.
Here is the perfect location where I have built a two level rhombic dodecahedron structure.
It is well used as a shelter for local wildlife and as a place of meditation.
There is a joy in being surrounded by the virtual matrix design of our universe.
It is well used as a shelter for local wildlife and as a place of meditation.
There is a joy in being surrounded by the virtual matrix design of our universe.
In order to enhance the energy field I am planning to sink copper rods into the ground next to each post. There will be twelve in all.
I will join long lengths of copper braided ribbon to each rod and then attach these ribbons, running from bottom to top, to the exterior framework of the structure. The twelve copper ribbons will be joined at the peak.
I will join long lengths of copper braided ribbon to each rod and then attach these ribbons, running from bottom to top, to the exterior framework of the structure. The twelve copper ribbons will be joined at the peak.
The final result.... a secret jungle retreat!
Saturday, July 5, 2008
Sphere Packing
bobhexa 11-26-2007, 09:18 PM
#1
Sphere packing
As a geometist I remain intrigued by the attached diagram which I drew a long while ago. I endevoured to draw eight circles around a larger circle so that all the circumferences touched. By rough measurement it appeared that the ratio of the diameters of the smaller circles to the diameter of the larger centre circle was very close to the Golden Section.A good friend and mathematician advised me that it was close but no coconuts were to be won.Well I have since wondered about the fact that it was so close. I have expanded my wondering to go 3 dimensional and to consider spheres instead of circles. So the question is gentlemen would spheres which are manufactured to have their diameters exactly a golden section ratio of the central sphere be able to pack around this central sphere and make an exact fit.
Location: Hawaii, Oahu, Manoa Valley
Bignose (Maths Expert) 11-27-2007, 01:33 PM
#2
The closest packing circles of equal radius can have is . The closest packing spheres can have is . You might really like the article "Cannonballs and Honeycombs" by Hales in Notices of the AMS, vol 47, 2000, because it has a neat discussion of these issues and other shapes being close packed.
Location: Iowa
bobhexa (Quark) 11-28-2007, 02:25 AM
#3
I thoroughly enjoyed your reference to Thomas Hales .'Tis a very good informative piece of work. Absorbing reading.I gradually understood that he was talking mainly about similar sized circles or similar sized spheres.My , I think, bottom line question is as follows. Supposing that you have a single sphere of 100 mm diameter and you have a plurality of smaller spheres with a diameter of only 62 mm (Golden Section ratio).....are you able to to distribute the smaller spheres around the surface area of the larger sphere and maintain perfect contacts between all neighbouring spheres. Is it a perfect fit?Having pondered this, I am led to the conclusion that a c60 pattern is the answer.You guys are the mathematicians. Do the ratios of the diameters of the larger sphere and the diameter of the 20 smaller spheres come anywhere close to the golden section ratio?
If you think you can do a thing or think you can't do a thing, you're right. -Henry Ford
Location: Hawaii, Oahu, Manoa Valley
bobhexa (Quark) 12-09-2007, 02:35 PM
#4
Up to my eyeballs.
My attached diagram doesn't seem to be valid so I shall upload the image again. Using the C60 Buckyball as a reference, I joined 20 polystyrene foam balls together with toothpicks. In other words each of the twenty hexagons of the C60 were replaced with spheres.Surprisingly the overall appearance is of a pentagonal dodecahedron. Could it be that the truncated icosahedron can also be described as a truncated pentagonal dodecahedron?Could somebody please work out the math as to what the diameter of the inner sphere might be if the smaller surrounding spheres had a diameter of 100mm.Thanks in anticipation.
Location: Hawaii, Oahu, Manoa Valley
bobhexa (Quark) 12-20-2007, 07:35 AM
#5
In the interim I have built a large model of a buckyball and obtained fairly precise measurements of the internal diameter...that is from the centres of the diametrically opposed hexagons.It would seem that the ratio of the diameter of the hexagon(flat to flat)to the inner sphere is certainly in the ballpark of the golden section. It is within hundredths of an inch.I am still wrestling with a method to calculate this mathematically.Any ideas Gentlemen.Mele Kalikimaka to all.
Location: Hawaii, Oahu, Manoa Valley
bobhexa (Quark) 12-29-2007, 04:22 PM
#6
In order to clarify I made up this model of 20 foam balls. They coincidentally fitted perfectly into this buckyball framework toy. Question....What would be the diameter of a sphere that fitted perfectly inside this model if the smaller surrounding spheres had a diameter of 100mm?
Location: Hawaii, Oahu, Manoa Valley
bobhexa (Quark) 03-29-2008, 11:26 PM
#7
Nobody seems to have applied their brains to this question
Is my description too ambiguous or is this a really difficult computation.I would be so grateful if somebody would please try to calculate this out. Aloha Bobhexa
Location: Hawaii, Oahu, Manoa Valley
cjohnso0 (Quark) 04-12-2008, 07:04 AM
#8
Well, on first try I came up with an maximum inner sphere size of ~180.2517mm.Basic methodology was as follows, refer to this page (Figures 2 and 2A) http://www.kjmaclean.com/Geometry/dodecahedron.htmlI took the 100mm sphere diameter and set this equal to the length of one of the dodecahedron sides. Using this, you can find the length of the sides of the cube which fits perfectly into the dodecahedron.I came up with cube side length of 50/cos 72.Next, I figured that the tetrahedron inside the cube would have the closest packed 4 spheres. So I found the length of it's sides, which came out to be 50*sqrt(2)/cos72.Using this, the radius of a sphere circumscribing the tetrahedron is r=sqrt(6)/4 * Side Length.Subtract 2 times the radius of the balls, or 100mm and I got my answer.Final equation was (hopefully)D = 2*(50*2*sqrt(3))/(4*cos 72) - 100which was ~180.2517Let me know if my method is way to confusing to read. Not that I'm 100% sure I'm even close....
Location: Western Mass.
bobhexa (Quark) 04-13-2008, 07:04 PM
#9
Getting closer
Dear cjohnso0Hi Aloha and thank you for your reply. The Kenneth MacLean site is a golden referral...I must buy his book.I was with you all the way through to the finding the length of the side of the cube. It was at that point I got lost but I realised that the diagonal of that cube coincides with the diametrically opposed spheres of my model dodecahedron.(see earlier Attachment) And then as you say subtract from this diagonal length twice the radius of the balls, or 100mm . Does that give us the same answer?Thanks again
Location: Hawaii, Oahu, Manoa Valley.
#1
Sphere packing
As a geometist I remain intrigued by the attached diagram which I drew a long while ago. I endevoured to draw eight circles around a larger circle so that all the circumferences touched. By rough measurement it appeared that the ratio of the diameters of the smaller circles to the diameter of the larger centre circle was very close to the Golden Section.A good friend and mathematician advised me that it was close but no coconuts were to be won.Well I have since wondered about the fact that it was so close. I have expanded my wondering to go 3 dimensional and to consider spheres instead of circles. So the question is gentlemen would spheres which are manufactured to have their diameters exactly a golden section ratio of the central sphere be able to pack around this central sphere and make an exact fit.Location: Hawaii, Oahu, Manoa Valley
Bignose (Maths Expert) 11-27-2007, 01:33 PM
#2
The closest packing circles of equal radius can have is . The closest packing spheres can have is . You might really like the article "Cannonballs and Honeycombs" by Hales in Notices of the AMS, vol 47, 2000, because it has a neat discussion of these issues and other shapes being close packed.
Location: Iowa
bobhexa (Quark) 11-28-2007, 02:25 AM
#3
I thoroughly enjoyed your reference to Thomas Hales .'Tis a very good informative piece of work. Absorbing reading.I gradually understood that he was talking mainly about similar sized circles or similar sized spheres.My , I think, bottom line question is as follows. Supposing that you have a single sphere of 100 mm diameter and you have a plurality of smaller spheres with a diameter of only 62 mm (Golden Section ratio).....are you able to to distribute the smaller spheres around the surface area of the larger sphere and maintain perfect contacts between all neighbouring spheres. Is it a perfect fit?Having pondered this, I am led to the conclusion that a c60 pattern is the answer.You guys are the mathematicians. Do the ratios of the diameters of the larger sphere and the diameter of the 20 smaller spheres come anywhere close to the golden section ratio?
If you think you can do a thing or think you can't do a thing, you're right. -Henry Ford
Location: Hawaii, Oahu, Manoa Valley
bobhexa (Quark) 12-09-2007, 02:35 PM
#4
Up to my eyeballs.
My attached diagram doesn't seem to be valid so I shall upload the image again. Using the C60 Buckyball as a reference, I joined 20 polystyrene foam balls together with toothpicks. In other words each of the twenty hexagons of the C60 were replaced with spheres.Surprisingly the overall appearance is of a pentagonal dodecahedron. Could it be that the truncated icosahedron can also be described as a truncated pentagonal dodecahedron?Could somebody please work out the math as to what the diameter of the inner sphere might be if the smaller surrounding spheres had a diameter of 100mm.Thanks in anticipation.
Location: Hawaii, Oahu, Manoa Valley
bobhexa (Quark) 12-20-2007, 07:35 AM
#5
In the interim I have built a large model of a buckyball and obtained fairly precise measurements of the internal diameter...that is from the centres of the diametrically opposed hexagons.It would seem that the ratio of the diameter of the hexagon(flat to flat)to the inner sphere is certainly in the ballpark of the golden section. It is within hundredths of an inch.I am still wrestling with a method to calculate this mathematically.Any ideas Gentlemen.Mele Kalikimaka to all.
Location: Hawaii, Oahu, Manoa Valley
bobhexa (Quark) 12-29-2007, 04:22 PM
#6
In order to clarify I made up this model of 20 foam balls. They coincidentally fitted perfectly into this buckyball framework toy. Question....What would be the diameter of a sphere that fitted perfectly inside this model if the smaller surrounding spheres had a diameter of 100mm?Location: Hawaii, Oahu, Manoa Valley
bobhexa (Quark) 03-29-2008, 11:26 PM
#7
Nobody seems to have applied their brains to this question
Is my description too ambiguous or is this a really difficult computation.I would be so grateful if somebody would please try to calculate this out. Aloha Bobhexa
Location: Hawaii, Oahu, Manoa Valley
cjohnso0 (Quark) 04-12-2008, 07:04 AM
#8
Well, on first try I came up with an maximum inner sphere size of ~180.2517mm.Basic methodology was as follows, refer to this page (Figures 2 and 2A) http://www.kjmaclean.com/Geometry/dodecahedron.htmlI took the 100mm sphere diameter and set this equal to the length of one of the dodecahedron sides. Using this, you can find the length of the sides of the cube which fits perfectly into the dodecahedron.I came up with cube side length of 50/cos 72.Next, I figured that the tetrahedron inside the cube would have the closest packed 4 spheres. So I found the length of it's sides, which came out to be 50*sqrt(2)/cos72.Using this, the radius of a sphere circumscribing the tetrahedron is r=sqrt(6)/4 * Side Length.Subtract 2 times the radius of the balls, or 100mm and I got my answer.Final equation was (hopefully)D = 2*(50*2*sqrt(3))/(4*cos 72) - 100which was ~180.2517Let me know if my method is way to confusing to read. Not that I'm 100% sure I'm even close....
Location: Western Mass.
bobhexa (Quark) 04-13-2008, 07:04 PM
#9
Getting closer
Dear cjohnso0Hi Aloha and thank you for your reply. The Kenneth MacLean site is a golden referral...I must buy his book.I was with you all the way through to the finding the length of the side of the cube. It was at that point I got lost but I realised that the diagonal of that cube coincides with the diametrically opposed spheres of my model dodecahedron.(see earlier Attachment) And then as you say subtract from this diagonal length twice the radius of the balls, or 100mm . Does that give us the same answer?Thanks again
Location: Hawaii, Oahu, Manoa Valley.
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