Expo 67 Dome - revisted

[Chris Kitrick]

This is an update regarding the geodesic geometry of the 250 foot diameter Montreal Expo 67 dome.

I am fortunate enough to be working from copies of the original blueprints that contain all the frame edge lengths thus reverse engineering the tessellation details.

The outer frame incorporates two tessellation methods as have been pointed out before. I have attached a diagram that illustrates the specifics. Both methods produce only isosceles triangles plus the one equilateral one in the center of the icosahedron face. The entire equator of the dome is equally divided into 80 angular segments (4.5 degrees). This evenly divided equator greatly simplifies the lesser circle progression downward from the equator to the ground.

The two tessellation methods result in 23 different member lengths for the outer frame from the equator up. I have labeled the methods by the classification described in my 1990 paper ("A Unified Approach to Class I,II,III Geodesic_Domes").

The inner frame is an interesting result of trying to limit the number of unique members. Each triangular face on the Class I outer frame extends a center vertex inward to a radius of 121.5 feet such that each inverted pyramid has three identically sized legs. Thus for every outer Class I triangle there is only one new edge length to reach the inner radius. The frame sizes of the inner hexagonal frame is the result of connecting the aforementioned inverted pyramids.

Cheers,

Chris

[Gerry in Quebec]

Chris and others,

 

Thanks for those details of the Biosphere, Chris. Your timing is good since this summer (July 1 to be precise) marks the 50th anniversary of the opening of Expo '67 and the 150th anniversary of Canadian Confederation. If memory serves from my teenage years, the American pavilion, now called the Biosphere, was the most popular in terms of public visits and certainly the most memorable.

 

***

Here are links to other Geodesic Help conversations about the Biosphere from past years, including photos, drawings and a video.

 

Drone video, 2015 discussion

https://groups.google.com/d/topic/geodesichelp/y9C_UupVYiE/discussion

 

Temcor-style subdivision, 2014 discussion

https://groups.google.com/d/topic/geodesichelp/ctWyVQMH4ao/discussion

(This is one of two methods used in the section of the dome above the equator.)

 

Subdivision described by Kenner in Geodesic Math, 2013 discussion

https://groups.google.com/d/topic/geodesichelp/JKJOEoo8f-4/discussion

(This is the second of the two methods used in the section of the dome above the equator.)

 

12 pix, 2011 discussion

https://groups.google.com/d/topic/geodesichelp/g260DA6di10/discussion

***

A few years ago I too used the 1966 architectural drawings of Fuller & Sadao to reverse-engineer the geometry. I've attached a pdf comprising several screen shots of the resulting computer model -- calculated in Excel, displayed in Antiview. Some of these show the "inverted pyramid" structure and inner hexagonal framework mentioned by Chris.

 

- Gerry in Québec

[Chris Kitrick]

Thanks for sharing all the additional links. Your model from the blueprints is excellent! I am glad someone has done the forensics on them to reconstruct the frame.

Since this is the 50th year of the opening are there any special events planned?

Cheers,

Chris

[borthelcash ']

The pattern of purple triangles and yellow pentagons seem to show the face arrangement of an icosidodecahedron if one extends the pattern below the equator.

Did the designers realize this when they were drawing up the blueprints?

[Chris Kitrick]

Bucky was very interested in great circles in the symmetry of geodesic domes from the earliest point so I would not be surprised that the engineers/architects involved were also. Bucky was more than happy to give lectures on geometry. None the less the Montreal dome tessellation was well suited to the project and its composition frame was based on one of the easier and well known methods before the advent of widespread computer use to do more complex trigonometry. (attached image source)

Cheers,

Chris

On Tuesday, June 20, 2017 at 1:29:49 PM UTC-7, Chris Kitrick wrote:

[Gerry in Quebec]

Chris,

I checked the Biosphere's website to see if any events are planned to mark the 50th anniversary of Expo 67 and the raising of the U.S. pavilion. At the following link there's an announcement about something called "Echo 67"  -- looks like a retrospective exhibit. But when I click on the link to get the details, I get an error message.

https://www.ec.gc.ca/biosphere/default.asp?lang=En&n=D2B51169-1

I will send an e-mail to Canada's federal environment ministry in the hope of getting more info.

Personally, I think it would be great if something could be done to honour architect Shoji Sadao, Bucky's partner, who made the project happen. Last I heard, he was splitting his time between the U.S. (NYC) and Japan. He turns (or turned) 90 this year.

https://en.wikipedia.org/wiki/Shoji_Sadao

- Gerry in Québec

[Chris Kitrick]

Just to finish up on the topic I have attached a color coded diagram of the unique triangles on the dual method used on the equator up section of the dome. You can clearly see the pattern of the methods as they diverge from the equatorial great circles running through the mid-edges of the icosahedron.

I look forward to seeing the dome again some day. Haven't seen it since 1968.

 

Cheers,

Chris

On Tuesday, June 20, 2017 at 1:29:49 PM UTC-7, Chris Kitrick wrote: