Using Mountain Ranges to Predict the Shape of the Earth

[Rory]

A small number of flat earth believers have devised experiments which utilise views of distant mountain ranges. The idea is that distances and elevations can be used to determine whether the earth is flat or a sphere. In some ways, it is something akin to a large scale Bedford Levels experiment, wherein peaks in the far distance can be compared to the apparent heights of peaks in between. It is also known by some as "forcing the line".

The following photograph was taken by a flat earth believer who thought he had discovered an 'impossible' view (as far as the spherical earth is concerned) of the Blue Ridge Mountains in North Carolina.


Source: https://www.youtube.com/watch?v=DPDtMQqlprk

While the conclusion that he came to was shown to be incorrect because of a mistake he made in discerning his elevation and location (thread here) the photo he took is still useful as far as figuring out the shape of the earth.

First of all, the eagle-eyed among you will have noticed that the top of the lookout tower is apparently higher than the summit of Pinnacle Mountain. Whereas, given his viewer observation height of 5481 feet, were the earth flat, he should be looking down at the lookout tower, and up towards Pinnacle.



Also, in addition to this, on the flat earth Graybeard would not appear lower than the lookout tower.

So the above photo disproves the flat earth theory that it sought to support - but how does the sphere fare, and how can we test it?

Well, perhaps one of the quickest ways to so do is to work out the angles that are formed between the viewer and the distant peaks. And for that we need the distances, the elevations, and a bit of trig, like so:


Source: I made this! With MS Paint! ;-)

What we would expect to find, were the photograph above to tally with the spherical earth, would be the following:

• The smallest viewing angle, because it's the highest in the photo, would be to the lookout tower
• Next would come Pinnacle
• Next would come Graybeard and Bald Knob, with fairly similar angles
• The largest viewing angle would be for Fryingpan
• Also, the difference between the angles to Fryingpan and Graybeard, and the difference between the angles to Pinnacle and the lookout tower, judging by the horizontal red lines above, should be roughly similar

Crunch some numbers - excel spreadsheet attached, containing the formulae - and what we find is:



Whaddya know? It works.

Meanwhile, on the flat earth...



The predicted order of apparent heights is completely out of whack.

Now to Picture #2, which was also used to try and prove the flat earth in a youtube video: this time, a photo of the Colorado Rockies taken from the summit of Snowmass, at 14,105 feet (including six feet for the camera).


Source: http://images.summitpost.org/original/766588.JPG

Again, we can immediately see that this simply doesn't work for the flat earth model. The viewer is looking up to Maroon Peak, but looking markedly down on Mount Shavano, which is higher than both Maroon Peak and the viewer, while Mount Harvard appears lower than Pyramid Peak, even though it's almost 500 feet higher.

Similarly, peaks that should appear to be at about the same apparent height - particularly Castle Peak, and the much more distant Mounts Yale and Shavano - clearly demonstrate a significant difference.

Once again, we see that viewing mountain ranges can be used as a very simple and effective means of testing whether the earth is flat. And, once again, we can use the information in the photo to test the spherical model.

First of all, I'll draw in a different line:


And from that make the following predictions:

• The mountains on or near the line will have very similar viewing angles
• Maroon Peak and Castle Peak will have the smallest angles
• South Hayden will have by far the largest viewing angle
• Of the peaks on or near the line, Pyramid Peak will have the smallest viewing angle

Once again, crunch the numbers and...



Not bad at all: it seems to work. All the viewing angles appear in the corresponding order to the summits in the photo, and at relatively similar intervals to their horizontal positions.

Notes:

• This will of course work for all mountain ranges everywhere, as long as we have an accurate elevation for the photographer
• Summit elevations vary from source to source. While this won't affect the conclusion, it will affect the precision of the calculations. (The source I used was the National Geodetic Survey)
• In addition to that, the angles calculated for a globe earth are not perfectly accurate - maybe about 0.01-0.02 degrees off - as I haven't taken into account the difference the 'tilt' would make. It's most likely negligible, but would be nice to have - so if anybody wants to add that to the spreadsheet and return it to me, please go ahead, and I'll update this post

In conclusion: photos of mountain ranges show that the earth cannot be flat, and support the model of the earth as a (more or less) 3959-mile radius spheroid.

 

[Rory]

Previous to my doing the above, Youtube user "Adams Truth Journey" [sic] had used the same technique to determine the shape of the earth, viewing mountains in Arizona from the Mogollon Rim Visitor Center:

Source: www.youtube.com/watch?v=iwN1ODgjq5s

In the video he visits the viewpoint, calculates what he believes should be the apparent order of the mountain peaks, and compares his calculations with what he sees in reality, coming to the conclusion that the results support a flat earth.

The reason for this, however, is that he is using "drop height" to work out the apparent order of the mountain peaks, rather than finding the viewing angles. As demonstrated below, this will result in an incorrect conclusion.

Consider the following photograph:



In reality, the chair in the foreground is higher than the table, the hedge, and the house nestled among the trees on the right, even though all three of those objects appear higher in the picture.

Using "drop" figures, though, as Adam did in his video, would mean that we would expect the chair to be highest in the photo, followed by the hedge, the table, and the house.

It's perhaps somewhat counterintuitive; but a simple profile diagram may make it easier to follow:



Using viewing angles, from smallest to largest, predicts an apparent height order of hedge > table > chair - which is what we see in reality.

Note: Viewing angles are required to work out the apparent height of an object on a hypothetical flat earth too: as in the above garden shot, wherein curvature is not an issue.

Back to Adam's view from the Mogollon Rim Visitor Center:



His photo is pretty poor - in fact, it's actually taken from the information board at the lookout - but gives us just about enough to make out his chosen peaks.

He then works out the drop heights and concludes that the apparent viewing order on a spherical earth should be as follows:


(8:51 in the video)

(Note: his distances for the two furthest mountains were about 4 miles out.)

He also calculate what he believes we should see on a flat earth - but, once again, because he's using drop rather than viewing angles, his figures aren't right.

Using the attached spreadsheet to calculate the viewing angles for both a flat and spherical earth returns the following:



Which, interestingly enough, supports both models, and contradicts neither - in apparent height order, at least: analysing angle intervals may reveal otherwise, though I'd say a better photo of the mountain range would be necessary for that.

 

[Rory]

Another photo I looked at applying these techniques to was a shot of Colorado's Front Range, taken from an aeroplane, and also used in a youtube flat earth video:

Source: http://www.pikespeakphoto.com/aerials/front_range.html

It's a great pic, and well suited to the method, but unfortunately only included vague information about where it was taken from, with the photographer saying it was shot "about 5 miles east of Larkspur" and from an unknown altitude estimated at around 15,000 feet.

Still, even without a definite altitude, I think it's an interesting picture to look at.

The first thing to notice is that Humboldt Peak, Pikes Peak, Kit Carson Mountain, and Challenger Point are all at approximately the same apparent heights, just a pixel or two apart.



Using this, we can say that, if the earth were flat, the camera would have been at around 14,080 feet, or pretty much level with Pikes Peak.

Meanwhile, for a spherical earth, working out the elevation where the camera would have been by calculating similar viewing angles to these peaks returns an altitude of around 16,285 feet, which is similar to where image overlaying on Google Earth puts it.

Looking at the flat earth model first, for this portion of the photo, there isn't a problem - the only mid-distance mountain is Pikes Peak, and all the other mountains are of sufficiently similar distances from the viewer for their apparent heights to appear correct.

Looking at the left-hand portion of the photograph, however, provides more variety in both peak heights and their distances from the camera, which should then provide enough information to find whether the viewing angles match the apparent height order.



Interestingly, once again, as far as apparent height order goes, neither model is significantly contradicted by what's in the photo - flat earth even passes the test for the sphere earth viewer altitude.



Conclusion: once again, the flat earth model will not always 'fail' these mountain range apparent height order tests. It seems that the mountain range needs to fulfil certain criteria in order to provide a useful result. I'm not a hundred percent sure what that criteria is. One thing that strikes me about this photo is that the distant mountains are all significantly higher than the nearer mountains - indeed, it seems that in this case the elevations and distances from the observer increase almost in tandem. The photos in the OP, however, contain mountains of similar elevations both near and far, as well as mountains of contrasting elevations. Care should be taken, therefore, in not drawing conclusions from analysis of just one mountain range.

PS I did mention in the OP that I hadn't factored in for tilt in my equations: for this photo I did do that, and after much laborious algebra and fannying around with equations in excel I discovered that...it makes basically no difference: an average of 0.002°.

 

[Rory]

My final attempt to extract something from this photograph was to consider the whole panorama and to add a scale to see just how well the predicted angles align with what we are seeing in the photograph:


(Scale shown is for sphere earth angles; hypothetical flat earth scale would be different.)

I can then compare the geometrically predicted viewing angles with the angles shown in the photograph:


(Selected representative peaks only)

The degree of accuracy is pretty remarkable, with only those mountains 120+ miles distant - Slide Mountain, and the group around Lindsey - appearing to show a slight discrepancy (I have no idea why this would be).

Testing the flat earth, meanwhile, also shows a good degree of accuracy when using the results from the hypothetical viewer elevation of 14,080 feet:



However when testing the data using the presumed correct viewer elevation of around 16,285 feet, I find the key to how to use these photographs:



Though the order of peaks is almost correct, many of the viewing angles are significantly off, and Pikes Peak stands out noticeably as "being in the wrong place": according to flat earth geometry, it should appear much lower in the photograph.

This illustrates the question I raised earlier, about necessary criteria: clearly, not having peaks of similar heights both near and far can lead to incorrect conclusions. As well as using only apparent height order, rather than closer analysis of viewing angles.

Naturally, this is all a very long-winded way of investigating something mostly pointless. But, if more "mountain range force the line proofs" do begin to surface, at least there's a resource looking at them, and showing how and why they both disprove a flat earth, and tally with a globe.

FINAL CONCLUSIONS:

• Though I say in the OP that this should work for all similar shots of mountain ranges, having looked at a few it's clear that the flat earth will 'work' for some, as far as apparent height order goes. Shots don't only require mountains of different heights and distances, but for the best results, mountains of similar heights significant distances apart
• For shots where the flat earth apparent height order is shown to match what is seen in the picture, looking more closely at the viewing angles will probably show discrepancies
• As far as height order and viewing angles goes, all of the above photos tally with a spherical earth - though not with a perfect degree of accuracy. Looking at these pictures is not a precise science. But, it seems, precise enough
• Some of these photos are an excellent disproof of the flat earth; others take a bit more work to do so
• An accurate and verifiable elevation for the viewer really is a necessity for best results
• The same goes for a good photograph wherein the peaks can be clearly discerned

 

[Rory]

I know I said I was done with this, but due to the overwhelming response, and continued demand/clamour for more of these, I figure one more wouldn't hurt.

This one is a shot from South Sister in Oregon, looking north into Washington. It's a good one because there's a nice variety of distances and elevations, and it's a very high-res and clear image.

First off, I'll visit peakfinder.org, put 'South Sister' as my location, and use that to figure out which peaks I'm looking at:



I'm going to take distance information from peakfinder (seems accurate enough, to within 0.1 miles, and saves me measuring it on google maps) but I'll take elevation information from NAVD88, which is apparently the most up to date and accurate (peakfinder elevations are sometimes quite significantly different).

Here's my picture:



And here's how the predicted viewing order for both spherical and flat earths compare with what we see in reality:



Two things to note there: number one, what we should see were the earth flat is completely at odds with what we actually see. This photo is probably an especially good proof that the earth isn't flat because of the variety in distances and elevations, and, in particular, because mountains much higher than both the viewer and nearby mountains are located in the far distance, where the earth's curve makes them appear substantially lower in the picture.

The second thing to note is that the globe earth doesn't quite pass the test either: Rainier and Middle Sister are switched. Looking more closely at the viewing angles perhaps sheds some light:



As we see, almost all the mountains are within a good range of error, save Rainier and Mount Adams. Probably not by coincidence, these two mountains are also the furthest from the viewer - 190 and 146 miles respectively. As in the previous image, it appears there may be some effect at play which causes peaks ~110+ miles away to be more liable to error than those closer. In all cases, the predicted angles are greater than that shown in the photograph, so there does appear to be a degree of consistency to the error.

I have no idea what is causing this. Other than to hazard a guess that the line of sight is curved due to refraction, and by the time it has travelled 110+ miles, it has done so sufficiently enough to create a +0.1° error.

For completion's sake, here are the viewing angles predicted by the flat earth model:



What this shows is that the flat earth model predicts the following:

• Mount Rainier would be the highest in the photo
• Mount Adams would come next, then Mount Hood. These two would be very slightly below Rainier, at about the same apparent height as one another (in actual fact, the order of these three is reversed, and the intervals substantially different)
• Mount Jefferson would be 4th highest, just a little above eye level (in the photo, it is the highest)
• North Sister is predicted to appear 0.81° beneath Mount Hood, and be the third highest in the photo
• Middle Sister and Three Fingered Jack are predicted to be the lowest two mountains, at pretty much the exact same apparent height (they're actually at noticeably different heights)

Or, to visualise, taking Jefferson and the nearest peak to be in the right place, the red lines show where the other five peaks should be:



Probably enough nails in this particular coffin now.

As ever, spreadsheet attached.

 

[Rory]

As in the previous image, it appears there may be some effect at play which causes peaks 110+ miles away to be more liable to error than those closer. I have no idea what is causing this, other than to hazard a guess that the line of sight is curved due to refraction, and by the time it has travelled 110+ miles, it has done so enough to create a +0.1° error.

I just thought I'd have a little play with that notion. I took the peaks from the South Sister, Front Range, and Snowmass photos, and plotted a graph of distances against errors:



There may be something there. That's 46 mountain peaks in total, and pretty much all the peaks returning the greatest errors are those 110+ miles away, with no peaks 110+ miles away not returning above average errors. But there's no smooth curve, which one might expect, and a few peaks in the 20-60 mile range that have above average errors too.

Something to think about, and certainly with regard to peaks in the far distance. But definitely not enough data to draw any conclusions from.

 

[Rory]

Y'all are probably wondering - on the edge of your seats wondering, yet simultaneously too shy to ask - how I estimate the apparent viewing angles in the photos. Well, now that I've got my system down, and since you ask so nicely, I'll tell you:

1. Note the y-coordinates (ie, pixel heights) of all the peaks in the photograph
2. Choose two of the closest, most clearly discernible peaks
3. Find the difference in their pixel heights
4. Find the difference in their geometrically predicted angles
5. Divide the two to find the number of degrees per pixel
6. From this, work out where zero degrees/eye level would be
7. From each peaks' pixel height, calculate the angle that it appears to be at in the photograph

Note: while I'm using the predicted angles as a starting point, it isn't quite 'circular logic' as, were there a flaw in the geometry, most likely only these two angles would align, and the rest would be thrown out. This is what tends to happen when we test the results for the flat earth.

A case in point, since I didn't do it for this photograph yet, is the one from the Blue Ridge Mountains which kicked the whole thing off. Following the above method returns the following results:



For this photograph in particular, it really is a remarkable degree of accuracy between the geometrically-derived angles, and those deduced from the photo.

One wouldn't wish to appear biased, though. Here it is for the flat earth model:



As we can see, though the points we've based our measurements are, of course, right, all the others are some way off, and as was shown in the OP, in completely the wrong order.

And using two different points to take measurements from (this time the summits of Fryingpan and Pinnacle):



There is simply no way to make these mountain range photos fit the flat earth model, and no way to show them to be incompatible with the sphere earth. Indeed, all results so far show a remarkable degree of accuracy, given the method.

 

[Rory]

You guys! Why didn't anybody tell me that I needed to add refraction to all these angle calculations? It's almost as if you hadn't read everything, or opened all the spreadsheets, or double-checked my findings.

 

[Mick West]

You guys! Why didn't anybody tell me that I needed to add refraction to all these angle calculations? It's almost as if you hadn't read everything, or opened all the spreadsheets, or double-checked my findings.

It’s the holidays! Maybe in the new year.

 

[Rory]

Oops. In the very first image in this thread I included the observer location but actually wrote in the coordinates for one of the mountains. Silly me!

Here's the image corrected:

 

[Rory]

I've recently been working on streamlining and improving the calculator used in this thread, and I think I've got everything working nicely now. One thing was that none of the above included refraction in the calculations: this answers the query in Post #6, as to why the more distant the mountain was, the greater the error. For example, Rainier from South Sister was showing in the photo at -0.97° compared to a calculator predicted angle of -1.14°. Now with refraction included, the calculator is predicting -0.95° - much more in line with the error range in nearer mountains.

This stands to reason, of course, that the error would be larger over greater distances - and only took me eighteen months to get around to fixing.

Here are the other mountains that were showing as +0.025° error in the above graph:



So they're mostly much better now, with the exception of Kineo and Zwischen, and maybe California. If I'm ever so inclined I might take a look at why those ones are standing out. But, as a whole, that's 48 peaks with only one returning more than a +0.1° variation from what might be expected, and there are several reasons why that might be. And it certainly answers the question posed above, all those months ago.

The other big change in the calculator is that it works out angles above and below eye level much more efficiently; so from here on in, negative angles are always below eye level and positive ones above - which wasn't always the case in above posts.

All the flat earth angles remain the same, by the way - that's always been a doddle to calculate. What a shame the world isn't flat! It would be much easier working on things like this.

I'll probably revisit the above photos and do a round up soon.

 

[Rory]

I've recently been working on streamlining and improving the calculator used in this thread

That link is now here:

https://www.metabunk.org/threads/calculator-for-viewing-angles.10586/