Is there a mathematical formula for the Gestalt Law of Proximity?

Question

I would like to know if there are any mathematical formulas that define the Gestalt Law of Proximity.

For example, in the picture above, at what distance of Y would the 4th square be perceived not a part of the group?

------ Update: I apologize for not giving enough details in the question earlier

1. The context is for on-screen elements, say on a website, so all 3D attributes, viewing angle can be discounted.
2. Assume the elements are identical to one another. Same colour, shape, size, border, etc...
3. The only differentiating factor is the PROXIMITY.

Answer 1

I don't believe there is a common mathematical formula for that. Especially with perception theories. It would heavily depend on

• your user's screen (if it's a digital application?);
• angle and physical distance of viewing your overall object;
• user's eye condition (someone with astigmatism and without correction lenses?);
• units of X and Y distances (in your example) and etc.

But if I had to 'guess' the answer to your question, depending on various conditions... it would probably simply be

Y > X

PS: Keep in mind, it's not just the distance (literal proximity) that grouping can be achieved with; but also with same/different values, colours, shapes and sizes, thicknesses, sounds, etc.

Hope this helps a little.

Answer 2

No, there isn't, and there never will be. Just test with your users to find out what works best. If there were equations for things like Gestalt Laws, designs would be programmed using AI by now.

Answer 3

This question got me thinking.

Your values x and y represent gaps which convey different meanings based on sizes relative to each other.

I'm sure there's not an exact, scientifically proven, and precise multiplier for y, but I started thinking about other web design elements that rely on relative sizes to express meaning.

The first thing that I thought of was headers. Their meaning is entirely defined by their sizes relative to each other, so these sizes have to differ enough for users to detect.

Looking at the W3C's defined default header styles, I figured the scale between each h* elements to range between 1.17 and 1.33, with the average of each size increase to be ~1.23.

So that's where I'd start with your visual design. Try adding 20-25% more spacing between groups, and see how that feels.

Ultimately, you're designing for humans, and human perception is not an exact science and varies from person to person. There will be people who can notice an extra 10% padding, and also users who fail to notice an extra 40%. It's all about trying to make it work for the majority, and providing additional assistance when possible (meaning, if testing proves this to be difficult for many users, you may consider a "high contrast mode" or similar setting that actually draws boundaries between implied groups).

Answer 4

It's an interesting question but perhaps strives too far from the practical in to the abstract.

We could try to figure out the ratio between X and Y at which the average person would draw the difderence. We'd have to compare the ratios of items vs whitespace, screen size and viewing distance, and all sorts of variables. But say we end up at a ratio of 1.2345 - would you actually use it?

If some see the difference at 1.1 and some at 1.5 would you go for the average and confuse about half the audience? And would you be willing to sacrifice sticking to a grid for it? Or would you play it safe and use a whole gridblock as a gap?

Answer 5

To answer your question I would refer to Von Restorff effect

"...also known as the "isolation effect", predicts that when multiple homogeneous stimuli are presented, the stimulus that differs from the rest is more likely to be remembered. The effect is based on the isolation paradigm, which refers to a distinctive feature of an item in a list that differs from the others by way of dimension. Such distinctiveness, leading to the von Restorff effect, can be generated from changing the meaningfulness or physical nature of the stimulus in some way, such as in size, shape, color, spacing and underlining"

A more in depth analysis of the Von Restorff effect is made in The subtlety of distinctiveness

"The isolation paradigm is one method for manipulating the variable of distinctiveness, and the isolation effect thus comes to be viewed as an instance of distinctiveness effects in memory. The intuitive explanation of the isolation effect in particular and distinctiveness effects in general is that the perceptual salience of the distinctive event attracts additional processing"

Hope this helps. Cheers!

Answer 6

No, but there is for Fitts' Law, which is well explained here: https://www.interaction-design.org/literature/book/the-glossary-of-human-computer-interaction/fitts-s-law

It's not the same thing, but might provide some helpful aside insight.

Answer 7

Some answers suggesting trial-and-error got me into action:

The spaces between the elements are 1/3 of the elements width. And the supposed group space is based on the golden ratio - an eye-pleasing phenomenon known for couple of thousands of years. If you want to translate into numbers, the space ratio here is 1:1.618 (thus 61.8% wider).
There is some irregularity in the placement, however it's difficult to say whether the two elements on the right form another group.

Let's make the spacing twice as big (200%): Something comes up, however not clear yet, I'd say.

Another attempt is to use the space width equal to the element's (average) width: Seems clear to me.

Last possibility would be a "missing element" (2 x element spacing + average element's width) to check whether the groups are not to far apart: Pretty obvious, perhaps not really feasible if space is at premium.

To summarise - should you not find any mathematical rules, you can use the examples as guidelines to establish your own rule for the design.

Answer 8

As the Gestalt Law of proximity is a perception theory we would assume that there is no way to formulate a calculus that would give exact values.

The proximity is in effect simply based on perception. Keep in mind that: " perception may be defined as a process by which individuals organize and interpret their sensory impressions in order to give meaning to their environment"

So perception is an individual unique feature as not all people see or perceive the same.

The only way to know that your design is putting proximity in good use is user testing.