Just because someone called it the golden ratio, doesn't mean it's ever really used. In fact it's very conspicuous by its absence in the standards of the modern world.
If you examine the aspect ratio of painting canvases, film sizes, standard photographic paper, screen sizes or paper sizes, you will find that none of them follow the golden rule. In fact there is no consistency whatsoever in aspect ratios in the modern world. They plot all over the graph.
If you frame a picture perfectly in your camera, no matter what the film or sensor size, there often isn't a single standard paper size that will fit the same aspect ratio. In fact photographic paper sizes are all over the map as you increment sizes. Even the 4K standard is a different aspect ratio than Ultra HD. 2K is different from HD. Movie screen aspect ratios are also quite varied. When you think about it, it's quite stupid, like incompetent CEOs on power trips.
Most HD Camcorder's viewfinders show a cropped version of what's recorded, so your HDTV shows more than you bargained for. What's wrong with those designers?
I sometimes wonder why most artists restrict their artistic freedom by fitting their paintings to a specific arbitrary canvas size. I often trim my photographic prints to the aspect ratio that fits the scene.
American standard paper sizes such as A at 8.5" x 11" = 1.294, while doubling to B at 11 x 17 = 1.545 isn't even close. The European standard aspect ratio for paper makes the most sense as it is fixed at 1.414, so doubling maintains the same aspect ratio, so every European paper size has the same aspect ratio. They don't have the misfit problem when they shrink a print by one size.
So I wouldn't worry too much about your project not meeting the "golden rule". Let's just say it met your platinum rule. It's called artistic license, so take advantage of it.
One thing to keep in mind before you decide whether to use the Golden Ratio is to look at it in terms of overall display size as well as the spacing between the elements. All the samples in your blog post are either 3x4 or 5x6, so even though you have the Golden Ratio spacing of 1.6x between the individual elements that overall size of the samples as a unit doesn't reflect the Golden Ratio. Try a test where you print out a sample that is NxN and uses a spacing of 1.6x between the individual elements (horizontal and vertical). Then not only would the Golden Rule be reflected in element spacing but in the overall display.
Your BADASS Display will be 16x16 when finished so if you do use the Golden Ratio as the spacing factor then it will be not only 1.6x between elements but 1.6x horizontal vs. vertical in the total display. That might heighten the visual impact (positively, I suspect) of the Golden Ratio.
I don't know whether this will influence your opinion or that of your test subjects, but I'd make that part of the test just to be sure before you decide.
@Disquisitioner: One thing to keep in mind before you decide whether to use the Golden Ratio is to look at it in terms of overall display size as well as the spacing between the elements.
I agree -- thsi is the sort of thing that has a ripple-on effect -- it's not sufficient to pick a rule and use if for just one element of the display -- you have to keep everything in harmony -- my mind is still churning with thoughts on all of this...
Engineers typically being horrible aesthetic designers - myself included - it is useful to default back to that ratio whenever there is an arbitrary question of style involved - it is hiding in plain sight within everything I have designed over the past 30 years.
I have never found it to give any particular performance benefit in optical designs, but I did once develop an optical reflector optimization technique that used genetic algorithms to perfect reflective surfaces that consistently spat out Nautilus shells in preference to the expected parabolas.
"Hiding in the design" is the spirit in which many of its practitioners have used it over the past few thousand years anyway. A nod to other educated people viewing the work. Worst-case using it makes me feel like I stand shoulder to shoulder with the likes of Vitruvius, Medieval Cathedral Masons and Leonardo for a few modern moments. The idea of some universal truth hidden in a cell phone housing that the Renaissance masters would "get" is rather appealing to me. We all like inside jokes.
Again, I only use the ratio myself when no other criteria drives a particular dimensional selection, but I definitely do enjoy getting to use it when I can - in fact I have it on speed-dial in my Pro/Engineer and Altium seats here. The best aesthetic designers I know definitely don't use it on a regular basis. I recall using it as a driving ratio for surface development in molded plastic housings on one particular project and it did not look good at all. There is excellent debunking content around the subject on the Skeptoid site btw. I believe the number actually has its own Fbook page too - or did a few years back anyway.
In a world where the bulk of the work of the readers here is thrown into the recycling bin just a few years after they complete it, it feels nice to at least for a while stand alongside cathedrals that have lasted a thousand years. Unfortunately most of the time today if one's engineering decisions are not immediately driven by regulatory code, cost and compelling structural and mechanical fit reasons then the chances are you are missing something important from the design brief in the first place. Around 50% of those medieval cathedral towers fell down on the first go-around remember.
Even today in modern physics many believe there is a strange harmonic numeric relationship between the fundamental constants of the universe, so maybe there is something special about numbers after all – we just haven't found what it is yet. God may well be man writ large, but he or she probably looks nothing like a giant cell phone.
If you take some measurements of important areas of the palm of your hand, you will find something interesting. Take the distance from the "large" knuckle of your index finger to the inner point where your thumb "diverges" from the rest of your hand. This is line A. Take the distance, again from said knuckle of the index finger, to the equivalent knuckle of the pinky finger. This is line B. The ratio B:A is awfully close to Phi (properly known as the golden section). I deliberately chose these proportions for comfortable "hand appeal" with a project oof mine, the BLiVIT.