Post on 20-Feb-2023
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