If we only lived in three dimensions, then we should expect higher precision of measurements to produce more accurate information about the objects we are taking measurements about at the original scale. This would be the expectation of a mathematical space of three dimensions (3-D) — more precise measurements result in more accurate models of those objects. However, nature does not always mesh nicely with mathematics, since adding more and more precision to measurements at one scale shifts us into another scale with different objects, say from our bodily organs to cells or to molecules. More importantly, the objects we can measure at one level, say our body at our level, are not the same objects we can measure at other levels, say the cellular level. Further, the 3-D volume of our body does not change at different levels of scale since our body at our level and at the cellular or molecular levels has essentially the same physical 3-D volume. There is a statement in physics that two objects cannot occupy the same physical space. So, how can these different objects exist in the same physical 3-D space?

Now, we do consider scale as part of our reality. We even consider it to be a continuum of reality, just not a necessary aspect to locate an object in our 3-D model of space. We also experience an object (say our body) at one level as appearing differently at different scales. Our bodies consist of cells and proteins, molecules and atoms. We don’t see these at our scale, but we do at much smaller scales. How can these different objects all occupy the same 3-D space? Don’t these objects exist at different levels of scale, even in the same 3-D volume? This means we should require the scale of the objects to locate them in space — now requiring 4 measurements to locate an object in space.

Consider the tip of your finger. Consider how you might specify the tip of your finger at different levels of scale:

• The tip on our visual level
• The tip at the bone level
• The tip at the skin surface level
• The tip on the cellular level
• The tip on the protein level
• The tip on the molecular level

Consider these points at different levels defining a line and not a point. We could determine the tip of our finger at each of these levels. We could use a 3-D reference frame at our visual scale level and ‘slide’ the frame down each level to measure the tip at that scale, changing only the scale of the reference frame (see attached image). Moving a 3-D frame of reference along a line not in this reference frame is essentially the definition of moving in another, in this case fourth, dimension. Connecting the points of the tip of your finger at these different levels of scale would determine a line with a length not in any of our normal three dimensions — a line in a fourth scale dimension.

Maybe the reason these different levels do not take up the same 3-D volume is that the objects take up different 3-D levels of a four-dimensional (4-D) space. If we use a 4-D model, then these levels all make better sense than a 3-D model. This would ease quite a few explanations of the many levels of reality we have discovered in the past several hundred years and indicate that a 3-D model of reality is not the best (or even adequate) model.

A final thought here: It appears that we have to ‘travel’ along scale to get to one or the other objects at a different scale level (see interactive website by Huang bothers [https://htwins.net/scale2], or the videos Powers of Ten and Cosmic Zoom). The question then becomes: What are we travelling through and how do we measure the distance we travel? For this last item, the video Powers of Ten provides a clue that scale measurements are not the same as our traditional three, thus providing a potential reason why we have not yet been able to incorporate scale into our current model of reality. This provides a new direction for science and potentially mathematics.