Expanding Our Universe:

Part I Expanding Space

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Copyright © 2010-2012 by Donald G. PalmerAll rights reserved

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Table of ContentsProlog............................................................................................................................................................ 3

1 Introduction .......................................................................................................................................... 4

Part I – Expanding Space............................................................................................................................... 5

1. Scale as a Fourth Continuum of Space................................................................................................. 5

1.1. Scale as a Requirement of Position............................................................................................... 5

1.2. The Question of Accuracy and Scale............................................................................................. 9

1.2.1. Three Dimensional Issues Measuring Across Scale.............................................................10

1.2.2. Accuracy, Scale and Our Measuring Tools ..........................................................................12

1.3. The Question of Scale as an Aspect of Space..............................................................................14

1.3.1. Four Dimensional Objects...................................................................................................14

1.3.2. Consistent Perspective of Scale – The Large and the Small................................................17

1.3.3. Changing Perspectives of Scale...........................................................................................19

1.3.4. Measuring Across Scale: A 2-Dimensional Analogy ............................................................21

1.4. Implications of Scale as the Fourth Dimension...........................................................................23

1.5. Part I Conclusion .........................................................................................................................24

Bibliography & References..........................................................................................................................25

1.1 References ..................................................................................................................................25

1.2 Bibliography ................................................................................................................................25

Note: All figures are depictions to help the reader understand the discussion and are not meant to be

accurate representations of the actual objects, distances, perspectives, etc.

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Prolog

Philosophy includes the study of how we view reality and how our thoughts influence our

concepts of reality. This level of study is not easily amenable to experiment and verification via

standard methods of science. At the same time, this level of study, this meta-level to science,

can have a far ranging influence on what we call reality and therefore what science studies. It

is, therefore, of critical importance to understand the limitations and biases which our view of

and our concepts of reality impose on any scientific study of that reality. The topic of this book

is about a specific instance where both mathematicians and scientists utilize and implicitly

accept a concept that limits and biases our perspective of reality.

To assist an understanding of the author’s perspective and this book, some important concepts

should be noted:

Our concept of ‘reality’ is always a model of that reality. At no time should we confuse

the model with the reality and believe we have entirely ‘captured’ or ‘understand’ that

reality. The history of science shows us that very different models of reality have

claimed to be reality – only to be later replaced with a new model (e.g. a flat world,

classical physics, Newtonian gravity, a 3-dimensional universe). Further, it could be said

that the adherence of proponents to a model – as actually being reality – hindered the

acceptance of the new model. It is very possible we are in such a situation today from

at least scientific and religio-political perspectives.

The tools we use to think about, to measure, and to model reality influence, bias, and

limit our models of that reality. The extent to which, in particular, our mathematical

tools bias and limit our models is a key aspect of this book and the ideas captured

herein. That the mathematical tools we use to measure reality could keep us from

understanding and measuring significant aspects of reality should be a lesson kept in

mind for future models of reality.

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1 Introduction

Our conception of the world around us has changed dramatically over the past few millennia.

Even in the last 200 years our conception of the world has changed significantly. Science has

astounded us with technology and a view of the universe that appears quite accurate – if

primarily from a mathematical perspective and lacking aspects of comprehension. So to

suggest that this science clings to a perspective of reality which is hindering its own growth

seems to fly in the face of the evidence. However this perspective is not simply about science,

but about the mathematical tools science uses to divine our world. This limiting perspective,

this implicit assumption, is that the basic mathematical tools science uses for this vast array of

theory and application are actually adequate to the task at hand. It is the enlarging of this

perspective with new mathematical tools which will allow science to expand in ways currently

unimagined. However the depth of our current bias needs to become explicit if we are to

believe in this expansion of perspectives.

This book presents three parts to our changing perspectives:

1. Expanding our view of what space is. This is a change in the way we model and

understand space, in particular our geometric model of space. It is not a change in how

we experience space, just in how we model and explain what we experience. This is not

a small change in perspective. However that we could be missing such a perspective

due to our modeling (and mathematical) tools indicates the importance of these tools in

our own thoughts and concepts of reality.

2. Expanding our view of the modeling and mathematical tools we use to measure objects

and space. As a key reason for why we have not already accounted for 1) above, we

consider how our mathematical measuring tools have limitations which have obscured

from us more ‘accurate’ models of space and bias us against the expanded view of 1).

This item is actually the key to the change in perspectives presented in this book, even if

1) appears the larger change.

3. Expanding our view of how space and time interact. By expanding space, the question

comes up as to whether we have also changed the perspective of space, time, and

space-time. This part attempts to address this concern.

If the history of science is any guide to the future, the expansion of our understanding of the

universe is not over. Realizing and making explicit the underlying limitations and biases of how

we conceive of reality can still be a powerful method of philosophy which can influence that

understanding.

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Part I – Expanding Space

1. Scale as a Fourth Continuum of Space

The size, or scale, of objects in our world is a commonplace part of our experience of space.

We experience objects smaller than ourselves, like a pin, and objects larger than ourselves, like

a building. Science has shown us very small atomic particles and very large stellar objects. And

we even understand the scale of objects in our world to be along some sort of continuum –

from the very small to our scale to the very large1.

We have not, however, included scale in our models of space or the universe. We still consider

the only possible continuums, the only dimensions, of physical experiential space to be length,

width, and height. We think these are the only dimensions needed to uniquely locate an object

in space and in which objects can travel. So our model of physical space remains as it has been

since at least the Greek times, three dimensional.

However, if scale is a true continuum of space, then it should be included in our model of space

as a fourth dimension. This statement would seem to contradict our current model, or

conception, of the universe as a 4-dimensional ‘Space-Time’. It is, rather than a contradiction,

an expansion and re-interpretation of our current geometric model of what is ‘space’. Existing

equations need not suddenly be tossed out; however the interpretation of several of them

might need some re-evaluation.

1.1. Scale as a Requirement of Position

To understand how basic this concept of scale is, consider specifying a point in space. We need

a reference system for this, usually thought of as a three-dimensional set of axes. One axis

exists for each dimension and locating a point in space requires one measurement made along

each axis. Our current paradigm is that three axes (& three measurements) are all we need to

locate a point in space (see Figure 1: from Wikipedia http://en.wikipedia.org/wiki/3-

dimensional).

1Check out: http://htwins.net/scale/, http://www.scaleofuniverse.com or Gott, J. Richard and Vanderbei, Robert J.

“Sizing Up the Universe: The Cosmos in Perspective”; 2010; National Geographic

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Figure 1: Three dimensional Cartesian coordinate system with the x-axis pointing towards the observer.

However, reality is not a theoretic geometric space. And for scientific work, we do not need to

identify an infinitesimally small point in some geometric space. We need to identify the

position or location of an object in ‘real’ space. And we do this through measuring distances

along some agreed upon form of practical geometric axes.

A common example of specifying a location in space is to use the lay-out of a city giving the

intersection of two streets and the floor above street-level: say the corner of South and 34th

streets and on the second floor. As the streets intersect at a corner, we are provided two

practical axes along the ground. The third axis is the floor above the ground. So we have South

Street, 34th Street, and the second floor as the three measurements. Such a system is useful for

specifying a room or the location of a person, all objects of essentially the same scale.

However, this is insufficient if we are specifying the position of a molecule that is part of the

surface of a pen sitting on a table in this second floor room at the corner of South and 34th

streets (see Figure 2). We believe that three dimensions are sufficient to define a theoretic

point. And for a three dimensional (3-D) geometric model of space, this would be correct.

However, to locate an object in reality, the scale of our three-dimensional reference system is

also required to identify such a location.

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Sout

hSt

reet

34th Street

Figure 2: Table in room on 2nd

floor of 34th

and South Street. Molecule of pen on Table.

This is what is meant by ‘the fourth dimension is scale’: We require the scale of our measuring

sticks in order to specify a spatial position in the universe. This means we require four

measurements and four axes, which also means the appropriate geometric model of our

‘normal’ space is actually four dimensional. This dimension is not hidden or beyond what we

already can ‘see’ with technology. It is simply not accounted for in a 3-D geometric model of

space. And most of our experiences do not extend far from the scale of our bodies, so we don’t

need to account for this dimension in our normal day to day activities.

Let us re-consider the figure of specifying the position of a molecule that is part of the surface

of a pen sitting on a table in this second floor room at the corner of South and 34th streets (see

Figure 2). In this figure, showing the position of the molecule is a series of expansions, or

magnifications, starting with the figure of the table. These expansions or magnifications occur

along a continuum we call scale. And since scale can be considered as a ‘continuum’, it is a

physical axis of reality along which we can build out these successive expansions or

magnifications. Placing these expansions along an axis, it might look something like Figure 3.

And this model of reality could be at least as appropriate as a 3-dimensional one.

If locating an object in real space requires us to specify the scale of the object and we can

provide this along an axis we can also represent, then haven’t we actually described an

additional axis of real space, which our models need to account for?

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Figure 3: Molecule of pen on Table in room on 2nd

floor of 34th

and South Street – Scale View.

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Lest we be too hasty in this pronouncement, there are two particular questions we must

address in considering the scale of an object: 1) Are measurements at different scales really a

‘new’ dimension or are they only a question of accuracy within our normal three dimensions?

2) Are there actual aspects of ‘real space’ which we experience and which we need to

accommodate via a continuum of scale in our geometric models of space? These two questions

will take us through the rest of this first section.

1.2. The Question of Accuracy and Scale

If defining the position of that molecule is only a question of accuracy, we should have no

trouble measuring the distance between them. But how can we measure the distance from a

street to a molecule? There is a large difference in the accuracy of a measurement, say in

meters, of the position of a street and for that of a molecule, say in nano-meters (meters * 10-

9). Can we compare measurements made with units at vastly different levels of scale? If we

use the same units, can we compare a measurement at our scale and the molecular scale? Can

we use the same level of accuracy using solar system measuring sticks – like astronomic units

(the distance of the earth from the sun) or even stellar measuring sticks, like light years? In

locating that molecule, is it allowable to simply ‘scale up’ our units from, say, 3.415 * 10^(-9),

using nano-meter measuring sticks, to an accuracy of 3.415 * 10^(-15) using light year yard

sticks?

The scale of our measuring sticks is crucial to defining a location in our universe. This

statement might seem trivial, but considering scale as another dimension is new. Scale is as

important as the measurement along that stick in one of our ‘normal’ three-dimensional

directions. Consider defining the position of the sun, of the galaxy, of our body, of a cell, of a

protein, and of a sub-atomic particle. In all cases the scale of our measuring sticks – the units of

measure of our reference system - is a necessary ingredient to defining the position of that

object in our universe. The three measurements of length, width, and height are insufficient of

themselves. We must know the scale of the object we are locating in order to determine its

position.

A key statement of this book is that scale is a fundamental aspect of our universe, which we

must account for as a ‘new’ dimension. We must specify the scale of our measurements in

order to locate a position in space. This appears so basic that it has been an unacknowledged

assumption of how we measure our universe.

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1.2.1. Three Dimensional Issues Measuring Across Scale

Consider how we would specify the distance between two objects in space, such as a book next

to the pen in that second floor room and that molecule on the surface of the pen. At the scale

of the book, it would seem that all we need is a very accurate measurement of the position of

the molecule. But does this help us determine the distance between these objects? Let us say

the book is 30 centimeters from the edge of the table. How would we measure the distance

from the book to a specific molecule? At the scale of the molecule, the edge of the book is also

a huge assortment of molecules and no longer a nicely delineated edge (see Figure 3).

Figure 3: Molecules that make up the edge of a page in a book.

Consider measurements at the scale of the molecule: If we attempt to locate the position of the

book at the scale of the molecule, it is no longer ‘a position’ but an entire 3-dimensional galaxy

containing billions of other molecules compared to the molecule of the pen. Say we consider

the ‘center of volume’ of the book as the position of the book. Shouldn’t we also do this with

the molecule – using the center of volume of the molecule?

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Note that what we are really attempting to do is remove the aspect of scale from our

measurements. We attempt to do this by collapsing the position of the book into a singular

point representing the book – and the same for the molecule. These points are, however,

abstractions of the position of the book and molecule. They are also not the only possible

abstracted points of these objects. We might, instead, use the center of mass of the book and

the molecule. The two – the center of mass and the center of volume – will only co-inside if the

mass of the object is evenly distributed throughout the object’s volume. Note also that if we do

use the ‘center of mass’ we must now account for an additional characteristic of these objects

that may have nothing to do with the three-dimensional position of them – the characteristic of

mass. So the center of mass automatically brings in an additional measurement beyond our

expected three dimensions.

So let us, for the time being, pass over the center of mass as an additional characteristic and

use the center of volume of the objects. Would this solve the problem and only require three

measurements? The problem, however, is two-fold.

First, we need a reference system. This system still needs a scale for the units of measure – for

the measuring sticks defining these centers of volume. And we also need to determine a ‘level

of accuracy’ for the units we decide upon. If our measurements need to handle the center of

volume of the molecule, then we need to use measurements that can accommodate this ‘level

of accuracy’. So by judicious selection of our measuring sticks to accommodate the ‘level of

accuracy’ of our measurements (maybe we use angstroms?) we think we have determined the

frame of reference.

This leads to the second problem: We need to ensure the same ‘level of accuracy’ for the center

of volume of the book and for the center of volume of the molecule. However, can we

determine the center of volume of the book to the same ‘level of accuracy’ of that of the

molecule? Where, at the ‘level of accuracy’ for the center of volume of a molecule, is the

surface of the book – defined by billions of molecules? Might we need to determine the center

of volume of every molecule in the book in order to determine the center of volume of the

entire book to this ‘level of accuracy’? But, how do we determine the precise center of volume

of a molecule? We might need to determine the center of volume of all the sub-atomic

particles that comprise the molecule – quite a task.

The attempt to abstract the position of the book to a singular point (the center of volume of the

book) has led to a very difficult process, requiring a very large set of measurements.

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(Heisenberg’s uncertainty principle2 may even make this impossible.) Rather than help us, it

has actually made things worse, since we now need to determine the position of the center of

volume of the book at an appropriate level of accuracy to accommodate the position of the

center of volume of the molecule (which may be beyond any capability). Plus we have

forgotten the position of the actual object by abstracting it to a single point and the

measurement has become a far more difficult process involving billions of measurements and

calculations. This has not helped us – especially from a practical perspective.

If we consider different levels of accuracy, for the book and the molecule, we run into another

problem: The answer to the question “Where is the center of volume of the book” can be

significantly different, depending if we consider accuracy at our scale (measure the volume in,

say, liters to 5 significant places), or at the molecular scale (measure the volume in nano-liters

to 5 significant places) or even the sub-atomic scale (measure the volume in cubic angstroms to

5 significant places). At our scale determining the volume of the book is not too difficult even

to a ‘reasonable’ level of accuracy (say 5 significant places in meters). But just determining the

volume of the book at the level of scale of molecules might be rather difficult since, as we

noticed before, the edge of the book, made up of billions of molecules, is no longer easily

defined. To make matters worse, time creeps into the equation, since these molecules are all

moving and so, at the accuracy of the molecule, ‘the edge’ changes over time.

So we must account for scale in our model of physical space. We have found that a single

three-dimensional reference frame using three same-scale measuring sticks is insufficient to

determine the distance between any two objects in our universe. We must introduce the scale

of the objects as a required aspect of specifying any object in space. This means we always

need four values to define an object in space and the concept of time has not yet been

considered. Which means scale is a fourth dimension required for locating an object in space.

1.2.2. Accuracy, Scale and Our Measuring Tools

Above, we addressed the importance of scale to our measurements and took an initial look at

the problems of measuring the location of objects at different scales. It was noted that

measurements involve a level of accuracy with an error term. The accuracy, or more

appropriately the error term, of a measurement at one level of scale might not be comparable

2“In quantum mechanics, the Heisenberg uncertainty principle states by precise inequalities that certain pairs of

physical properties, such as position and momentum, cannot be simultaneously known to arbitrarily high

precision.” From Wikipedia http://en.wikipedia.org/wiki/Uncertainty_principle

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to a measurement at a different level of scale. The error term of the ‘larger’ measurement can

be much greater than the entire quantity of the ‘smaller’ measurement.

For example we might have a measurement of 2.15 meters, which is really 2.15 ± .005 meters

and another measurement of 3.41 * 10^-9 ± .01 (* 10^-9 meters). The potential error in the

first measurement (± .005 meters) dwarfs any accuracy of the second measurement at

0.00000000341 meters. Even if this first measurement were only off by .0001, this would still

be an amount 1 * 10^5 times larger than the second measurement (this is like the distance

from Boston to Washington, DC at our scale).

The question now becomes: Is measurement accuracy an inherent issue of any measurements

which cross scale? If this is the case, we will be unable to properly deal with measurements

across scale, since we will always be lacking considerable accuracy at the extremes of scale. At

first glance this might seem to be an inherent limitation of our abilities to measure. However,

this book suggests the problem resides with our implicit assumption that scale is simply an

aspect of being ‘more accurate’ and not, therefore, a real or measureable property of space.

This implicit assumption essentially results in a confusion of accuracy with scale hiding the idea

that scale could possibly be a measureable property of space.

This confusion can be overcome. We can consider scale to be a dimensional aspect of space

and hence work differently than simply being ‘more accurate’. By considering scale as a

measureable aspect of space, it becomes a dimension of space and a required aspect of

locating an object in space. Now, if we are able to measure scale, these measurements will

include their own accuracy of measurements ‘along’ scale. This means accuracy and scale are

not synonymous and should not be confused. Note that this conclusion is a result of simply

allowing scale to be in any way measureable.

If we wish to continue to hold a 3-D perspective of space and not allow scale as a position of

location nor as a measureable aspect of space, we should ask the question: If we can put scale

along a continuum, along a line with reference points (eg. 10^2 meters, 10^1 meters, 10^-5

meters, etc.) why is scale not measureable?

Let us consider a means of ‘measuring’ scale. Let us forget the possibility of scale being an

aspect of space – yet still allow it can be measured. We should still, therefore, designate scale

as having an axis along which we can measure. We need to use a different approach than our

other three axes, however, since scale applies to the measurements of all three axes. Note that

we always take the same units of measure for all three axes. We don’t measure a position

using length and height in centimeters and width in miles. So we could take the unit of

measure of scale as the unit of measure of this new axis – as the power of ten of the unit of

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measure for this axis (see Meter numbers on Title Page Figure). So let us consider the power of

ten of the units of measure for measurements in scale – as the measure along some ephemeral

scale axis.

This would mean a centimeter (1 * 10^-2 meters) is two scale units away from a meter and a

kilometer (1 * 10^3 meters) is three scale units away from a meter, although in opposite

directions. A curious question can now be asked: How ‘far apart’ is a centimeter and a

kilometer? In a strict three dimensional world where the difference between a kilometer and a

centimeter is see as differences in accuracy, this question seems inappropriate – not to apply.

In a four dimensional scale world, the answer is 5 scale units.

Doesn’t this ‘ephemeral’ axis provide a means of measuring scale? If so, then we have found a

means of measuring scale, which will need to include an accuracy – an error term – for

measurements in scale. This automatically divorces the concept of accuracy from scale and

(strongly) suggests scale is a measureable aspect of location and possibly of space. From this

discussion, alone, we might consider scale as an aspect of space. The earlier discussions in this

Part might still be necessary to consider it a dimension of space.

Note, also, that this measurement will appear to us as an ‘exponential’ axis. Each unit of ‘scale

measure’ is a power of ten in our traditional units. So 2 ‘units’ of scale will equate to

measurements 100 times larger in traditional units. This point will become of significant

importance later on.

1.3. The Question of Scale as an Aspect of Space

As noted at the beginning of this section, we do consider scale as part of our reality. We even

consider it to be a continuum of reality, if not part of our 3-D model of space. However, if we

are using a 3-D geometric model of space, does this model account for scale as a continuum,

separate from accuracy? If scale is an aspect of the space we experience around us, shouldn’t it

be included in our model of space? Are three dimensions really sufficient to capture this aspect

of our experiences of space?

1.3.1. Four Dimensional Objects

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 on the cellular level

The tip on the protein level

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The tip on the molecular level

What if you considered all these levels to specify the tip of your finger? Do you feel like all

these levels constitute the same point? Or are they different points connected by some sort of

line? Which of these seems the correct interpretation, forgetting our three-dimensional bias?

Consider these points defining a line and not a point. We should be able to determine the tip of

our finger at each of these levels. We could use a three-dimensional reference frame at our

visual scale level and ‘slide’ it down to each level to measure the tip at that scale, changing the

scale of the reference frame (see Figure 4). Moving a three-dimensional 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.

Figure 4: Finger at different scales.

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There is a well-known four dimensional object, called a tesseract. This is essentially a box

extended into four dimensions. Construction of a tesseract (see Figure 5: from Wikipedia

http://en.wikipedia.org/wiki/Tesseract):

Figure 5: A diagram showing how to create a tesseract from a point.

0 – A point is a hypercube of dimension zero.

1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit

hypercube of dimension one.

2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps

out a 2-dimensional square.

3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on,

it will generate a 3-dimensional cube.

4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-

dimensional unit hypercube (a unit tesseract).

Figure 6: Picture of a tesseract projected onto three dimensions (from Wikipedia).

(End of Wikipedia insert)

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Now, consider both the construction of a tesseract and the tesseract projection in relation to

the discussion of the tip of your finger. Assume your finger is a three-dimensional object at our

visual level and ‘move’ it in the 4-dimensional direction of scale. Wouldn’t a reasonable view of

this be a four-dimensional finger that exists at different levels of scale? And, if we smooshed

(note the technical term) these levels onto a 2-dimensaional visual level – might a picture of

your finger at these levels appear like a finger embedded in a finger?

Consider the tesseract (see Figure 6). What would a three-dimensional cube existing at our

visual level and, say, at the molecular level look like? If we connect the edges of the cubes at

these two levels, wouldn’t this appear like the cube-embedded-in-a-cube picture of a tesseract

above?

There is an interesting feature of the tesseract: The ‘inside’ cube of the tesseract projection is

actually another ‘lower’ surface of the 4-dimensional tesseract. Everything that appears to be

‘inside’ of this lower cube is actually outside of the tesseract. If the fourth dimension is scale,

this leads to a surprising prediction – that we have a ‘lower’ scale 3-dimensional surface to our

bodies and potentially all objects. This is very different than our current conception – that our

bodies and all objects extend indefinitely into the realm of the smaller. And if a particle much

smaller than this surface whizzed by (say a neutrino), we would expect it to miss this surface.

1.3.2. Consistent Perspective of Scale – The Large and the Small

If scale is the fourth dimension, what differences would we expect, compared to three

dimensions, when looking at objects at different scales? In three dimensions, looking at objects

which are ‘far away’ should be consistent no matter which direction we look. Consider how we

perceive very small and very large objects and the statements: “Molecules are too small to

see.” and: “Galaxies are too small to see.” Since there should be consistency to how ‘far away

objects’ appear, these statements suggest that both very small and very large objects are ‘far

away’. From our perspective, both objects are too ‘small’ to see. On the surface, this could

seem to support a scale-dimensional paradigm.

However, we might counter that we ‘know’ that galaxies are huge and that it is the distance

from us that causes them to appear ‘small.’ And we ‘know’ that molecules are tiny and that is

why they appear small. So it is not that molecules appear small, they are small. Plus, you might

add, we use a telescope to see large objects, while we use a microscope to see tiny objects. So

we don’t use the same tools to see each.

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The response to this counter argument begins by noting a three-dimensional world should

perceive all distant objects the same, requiring the same tool to perceive them. If we believe in

a three-dimensional universe, then we should perceive all far away objects in a consistent

manner requiring the same tool, say a telescope. Now, a distant galaxy doesn’t overwhelm us

with its size. From our perspective, it is so tiny that we must use a telescope to magnify it many

times in order for us to see it. However we now note that essentially the same is true of a

molecule and we must use a microscope to magnify it many times so it is large enough for us to

see. From a three-dimensional perspective we might argue only the large objects are far away,

which is why we use a different tool for the small objects. But both situations are essentially

the same, since we must magnify both a ‘large’ and a ‘small’ object many times for us to see it.

What accounts for this similarity in perceiving large and small objects – in a three-dimensional

paradigm? In a four-dimensional scale paradigm, we are just looking in two different directions

(requiring slightly different tools) and, in both cases; we are looking far away (since we are

always magnifying the objects large or small).

So a key question becomes: Why must we use ‘slightly’ different tools, even though they

perform essentially the same function? In a three-dimensional universe, why should we even

need a magnifying tool to see objects very close to us? A four-dimensional universe, however,

provides a reason for needing these different tools: There are two different directions we must

look in. The large and the small are not the same and the need for slightly different tools,

which work in a similar manner, is an expected consequence. One tool is needed to magnify

and see larger objects and another tool is needed to magnify and see smaller objects. This

need for different tools would, however be dependent upon what scale we are looking from.

This brings out an important predictive aspect of scale as the fourth dimension of our universe:

The tools we require to perceive other objects depend upon the scale we start from. We,

human beings, are not able to move in this dimension in the same way as our other three

dimensions (note than ‘up’ and ‘down’ are not as easy as left or right for us either). And since

we cannot easily move in this dimension, our perception of the universe is always from our

level of scale. To us, we are at a ‘normal’ scale, while the sun is huge. To see another person

from a distance (at our ‘normal’ scale), we would use a telescope. To see objects much smaller

than us, say cells, we would need a microscope. In a four-dimensional universe, from the scale

of the sun another sun would be at a ‘normal’ scale, while we would be tiny. From this scale,

we would need a telescope to see another sun; however we would need a microscope to view a

person on the surface of the earth, not a telescope. This is the prediction of a four-dimensional

universe: The use of different tools to see across scale would apply to the relative level of scale

we start from. If we change the level of scale, we need to shift which tools we use also. This is

not a prediction for a 3-dimensional universe.

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And what of a molecule in that other galaxy? Is it ‘too small to see’ or ‘too far away?’ We have

to admit that it is ‘too far away’ just like the galaxy itself. Would we need both a telescope and

a microscope to see that other-galaxy molecule? If everything is strictly 3-dimensional and we

don’t need a scale measurement, we shouldn’t need both. If the molecule simply occupies a

spot in 3-dimensional space – is only just very far away, all we should need is a telescope to see

it. But is this the situation we find ourselves in? A telescope pointed at the sun or moon

doesn’t allow us to see molecules in these objects. However, this should be the situation if

these objects are simply ‘too far away’. The requirement for both a telescope and a microscope

to perceive molecules in a distant sun is much easier explained in a four-dimensional universe

where scale is a required location identifier: We need a telescope to see objects at or larger

than our relative scale and we need a microscope to see objects smaller than our relative scale.

Again, scale is crucial to our perception and understanding of objects in the universe. Our

current three dimensions are insufficient to address this issue. We need a fourth aspect of the

universe, scale, to account for our perceptions of the universe. So we need tools to see across

this dimension – to see objects at a distant scale relative to ourselves (see Title Page Figure).

This explains why we would need both a telescope and a microscope to see that molecule in

that distant galaxy. We look in one direction of scale with a telescope, to see the distant galaxy,

and we need another tool, a microscope, to see in the other direction of scale to see the distant

molecule in that distant galaxy. Since the scale of an object is not one of our three axes to

locate an object, a strictly three-dimensional view of the universe does not provide this

explanation.

1.3.3. Changing Perspectives of Scale

The previous discussion also introduces a change in what we consider to be ‘close-by’. To say a

molecule of our finger is simply tiny presumes that molecule is ‘close by.’ In fact, don’t we

consider this molecule to be ‘part of’ ourselves? So it is not just ‘close-by’ but exists as part of

one of our cells, which is a part of us and so the distance should be zero. In a strict 3-

dimensional space world, we consider this tiny molecule just a more ‘accurate’ measurement of

a part of ourselves.

We should be able to test this situation, but how can we measure the distance from us to a

molecule? How can an object be zero distance from us, yet not be visible to us? We are now in

a similar situation to the one in section 1.2 above, when we considered measuring across scale.

And now the question comes up: Is there really zero distance between our level and that of the

molecule? If there is no distance, why do we need a microscope to even perceive a molecule if

it so close to us?

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Let us consider what it means to have distance between one point and another. From a strictly

geometric perspective, there only has to be space between the two points. But in our world,

traveling between two points, say New York and Los Angeles, means things change as we travel.

We don’t just see ‘space’ we see lots of other objects between the two points. In a three-

dimensional world, these objects would all also be three-dimensional objects (cars, houses,

buildings, pastures, fields, etc.). If there is no distance between two objects, there should not

be any other objects ‘between’ them.

There are some photographers who synthesize a picture from individual pictures at high

magnification (see image at http://www.docbert.org/MP/). We can look at the entire scene,

and then zoom in on one part again and again. This would seem to be what we should expect

in a 3-dimensional universe where ‘zooming in’ only changes the accuracy of our view. In these

images, the details are enhanced, but the overall perspective of what we see ‘zoomed out’ isn’t

different than what we see ‘zoomed in’. The objects in the picture do not change as we change

the accuracy of our view.

Taking ourselves and this molecule of our finger, is there no change between them? If we start

at our level and consider what would occur if we ‘travelled’ to the level of the molecule, what

would we see? We would see the ridges of our fingerprint, and the bumps in our skin. The cells

that make up our skin would come into view, followed by the one cell of which the molecule is

a part (see Figure 4). We would see the internal parts of a cell and then a particular protein of

which the molecule is a part.

Does this mimic the magnification of the high resolution photographs? Does this sound like we

are looking at the same point with no distance between us and the molecule? Or does this

seem like traveling across some distance, where the view changes? It is hard to rectify the

concept of a three-dimensional world that considers a molecule of our finger as simply needing

more ‘accurate’ measurements compared to our finger – and this sense of travel, of changing

perspectives and different objects, in the ‘shift of accuracy’ from our finger to that molecule of

our finger.

The concept of scale as a fourth dimension overcomes this problem, since this conception

requires we travel a distance to get from our finger (at our scale) to that molecule ‘in’ our

finger. The expectation of different objects and views as we travel this fourth-dimensional

distance is inherent in this conception – and is at odds with a strict three-dimensional

conception.

And the four-dimensional conception implies there is a distance between us, at our level, and

that molecule, at a different level. So that molecule is not ‘zero’ distance from us at our level.

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It can still be a part of us, just at a scale distance from us. This is not easily explained using just

three-dimensions.

1.3.4. Measuring Across Scale: A 2-Dimensional Analogy

Returning to the situation of attempting to measure the distance between objects at different

scales: Consider the edge of a book, at the meter scale and a molecule of the book, both

located on the axis we call ‘length’. The distance along the ‘length’ axis for one object might be

2.15 meters to the edge of the book, while the other might be 3.41 * 10^-9 meters further

along that axis, or 2.15 meters plus 3.41 * 10^-9 meters. The distance between these two

objects would appear to be the difference between their two measurements being 3.41 * 10^-9

meters.

However the tools we are using to make these two measurements, including our decimal

numeric system (and ‘scientific notation’ of exponents), are not adequate for performing these

calculations. We might think we are able to present the distance to the molecule as a single

decimal value of 2.15000000341, however we do not have 12 significant digits for the meter

measurement of the molecule. We must, therefore, actually break the distance into two, one

at each scale level: 2.15 meters plus 3.41 * 10^-9 meters – each with three significant digits.

As stated previously, one reason for this is that our measurements include an ‘error’ term. The

measurement of 2.15 meters is (for this example) really 2.15 ± .005 meters and the other

measurement is 3.41 * 10^-9 ± .01 * 10^-9 meters. The potential error in the first

measurement dwarfs any accuracy of the second measurement. Even if this first measurement

were only off by .0001, this would still be an amount 1 * 10^5 meters larger than the second

measurement.

Note that the attempt to consider the position of the molecule as 2.15000000341 presumes an

important item: That scale is not an important aspect of the location of an object in our world

and so we can just combine the two measurements. If scale is an important aspect, we would

understand that the two measurements are not along the same axis. A four dimensional (4-D)

scale perspective would presume that being at different scales means the ‘length’ axis of the

book and of the molecule are different lines, with a distance between them.

Consider if we believed we lived in a two dimensional (2-D) ‘Flatland’3 world, but that we really

exist in a three dimensional (3-D) world. What is ‘small’ actually exists above the plane of our

Flatland world. If we measure the distance to a Flatland book and to the position of a particle

3Abbott, Edwin A.; Flatland: A Romance of Many Dimensions; 1884

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only in the Flatland plane that makes up the book, we would only be measuring the distance in

our Flatland plane and not be accounting for its distance above that plane (see Figure 7).

2-D Table

2- D Book

Flatland

2-D ShadowPosition ofMolecule

3-D Position ofMolecule

Smaller ScaleAxis: ‘Z’

Z

Y’

X

Y

2-D Book

Figure 7: Flatland that is actually 3-D.

So if we believe in a 2-D world, we will be convinced that the molecule exists inside the 2-D

boundary of the book and that our (we believe) 2-D microscope magnifies the position within

the 2-D book. However, since our world is really 3-D, what we think is the position of the

molecule is really the 2-D ‘shadow’ position of the molecule. The actual position of the

molecule exists a distance ‘above’ the 2-D plane of Flatland and the 2-D book. And, since all

objects are actually 3-D objects, our microscope is really a 3-D microscope, which magnifies the

molecule at the real 3-D position of the molecule inside the 3-D book. We just don’t think it

does this and there is nothing obvious to convince us otherwise.

Now, the distance from the edge of the 2-D book to the 3-D position of the molecule is greater

than what we calculate from the edge of the 2-D book boundary to the shadow 2-D position of

the molecule. How we might detect this difference would be an interesting question. One

method might be observing that, if the shadow position is not the true position, then we should

find evidence of this distance. If we believe our world is 2-D and it is really 3-D, what might we

see if we look at the same 2-D shadow point at different scale levels between the meter scale

and the molecule scale? We might see different objects at different scales, since there is 3-D

space for objects other than the molecule to exist at this point. Isn’t this what we observe in

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what we believe is our 3-D world? Looking at the same position at different scales we observe

different objects.

An additional question can also be asked, if we consider the ‘distance from the edge of the

book’ using the 3-D boundary rather than the 2-D boundary: Which point of the 3-D boundary

should be used to measure from? There is a much larger boundary involved with the 3-D

boundary than the 2-D boundary and in a sense the 3-D boundary can be seen as being the

extension of the 2-D boundary into the up-down dimension.

So, if we really live in a 4-D scale world, that molecule of the (we believe) 3-D book is not inside

the boundary of the 3-D book, but at a distance ‘below’ the 3-D book – meaning at a different

scale - within the boundary of the actual 4-D book. And we should expect to observe objects

between the different scales, when looking at the same 3-D position.

1.4. Implications of Scale as the Fourth Dimension

This change to how we view the world, this paradigm shift, impacts how we think about the

universe. The question also arises: How does this impact what we do and where might this

change be used in our lives? This section cannot address more than a quick idea of how scale

as the fourth-dimension would impact our world.

Consider the medical imaging devices we have today. These image our bodies at a certain

scale, generally at that of our internal organs. Consider if we could image our bodies at smaller

scales, say that of blood vessels or of cells, and then build a whole-istic view of our body at all

these different levels. Since each level would be a three-dimensional image of our body, we

would need to build a four-dimensional image of our body to encompass all these levels. We

could then identify, through actual observation, how changes to a cell impact a particular

organ. There is a young discipline called ‘Multi-Scale Modeling’ which is attempting to

synthesize models at different scales. This discipline could be helped considerably if it used a

four-dimensional scale model.

Consider how to detect stress in metal, which involves changes in the structure of the metals’

molecules at specific points impacting the larger scale surface and integrity of the metal.

Combining models of the metal molecules (in a three-dimensional space) with measurements

of the metal at our level (another three-dimensional space) involves building a four-

dimensional model of the metal object.

Using only three-dimensional models, we will be unable to construct these multi-scale views of

our bodies and objects. This could impact many areas of science and technology.

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1.5. Part I Conclusion

The conclusion of this part is that the concept of scale as the fourth dimension is consistent

with our current experiences of the world. This ‘new’ dimension has always been there. No

hypothetical unseen dimension is required. This is simply a re-interpretation of our existing

experiences and noting that we must account for scale in measurements between objects. It

also requires us to separate accuracy from scale.

This concept can explain the problem we have of pin-pointing two very different sized objects

in space and, in part, of measuring a distance between them. The description of four-

dimensional objects, as with the tesseract, fits this concept of us existing in a four-dimensional

universe. One piece of evidence is the need for perceiving objects in different ‘scale’ directions

requiring different magnifying tools. Another piece of evidence is that we perceive different

objects as we ‘travel’ in scale along the ‘same point’ (e.g. the tip of our finger) – something that

‘increased accuracy’ doesn’t explain.

Considering our finger and a molecule in a cell of our finger, the two are not ‘zero’ distance

apart. We need a microscope to see across the distance separating our finger (at our scale) and

the molecule. Plus there are objects at scales in between these two, such as cells and proteins.

The strangeness is explained if we consider the molecule to be both at a distance from our scale

and as a part of us. This would mean we exist as beings that cross scale – as four-dimensional

beings. And just as there is a length to our arms and legs, so too is there a ‘length’, a measure,

to us across scale. And we should expect to have a ‘lower’ scale surface to our bodies.

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Bibliography & References

1.1 References

1. ClipArt from: http://classroom.clipart.com

2. Wikipedia on:

a. 3-Dimensions: http://en.wikipedia.org/wiki/3-dimensional

b. Tesseract: http://en.wikipedia.org/wiki/Tesseract

c. Heisenberg uncertainty principle: http://en.wikipedia.org/wiki/Uncertainty_principle

3. http://htwins.net/scale/

4. http://www.scaleofuniverse.com

5. High Resolution Images: http://www.docbert.org/MP/

1.2 Bibliography

1. Abbott, Edwin A.; “Flatland: A Romance of Many Dimensions”; 1884

2. Akhundov, M.D.; “Conceptions of Space and Time: Sources, Evolution, Directions”

MIT Press, Cambridge; 1986

3. Gott, J. Richard and Vanderbei, Robert J. “Sizing Up the Universe: The Cosmos in Perspective”;

2010; National Geographic

4. Parker, Sybil P. ed. In chief; “McGraw-Hill Concise Encyclopedia of Science & Technology”;

McGraw-Hill; NYC; 1992; 3rd. Edition.

5. Ray, Christopher; “Time, Space and Philosophy”

Routledge, NYC; 1991 [pgs. 1-79]

6. Wick, David; “The Infamous Boundary; Seven Decades of Controversy in Quantum Physics”;

Birkhauser Press, Boston; 1995