Check out the latest new Math book on the Fourth Dimension! The Fourth Dimension is the mysterious dimension which cannot be seen. Check out also our previous post on the Fourth Dimension Explained.
Mathematics popularizer Matt Parker, an Australian based in England, is a self-proclaimed “standup mathematician” perhaps best known for his numerous contributions to the Numberphile YouTube channel. He is also the Public Engagement in Mathematics Fellow at Queen Mary, University of London, and his new book, Things to Make and Do in the Fourth Dimension, is an ambitious and delightful addition to the current age’s plethora of high-quality volumes on recreational mathematics—even if most of the material he covers is focused on 2-D and 3-D. Like the extensive writings of legendaryScientific American columnist Martin Gardner this book seeks to make mathematics come alive for an intelligent and curious audience by engaging the reader in a lively informal style, and with irresistible invocations to roll up one’s sleeve and experiment. Parker also enlivens his chapters with numerous surprises.
Source: http://www.scientificamerican.com/article/how-to-get-to-the-fourth-dimension/?WT.mc_id=SA_WR_20150805
by Professor Bartle
-almost everywhere if there exists a subset 
with 
such that the proposition holds on the complement of N. 
, which is discussed in a 
are said to be equal 
when 
, for some 
with 
, 
, 
, there is a unique map ![p:[a,b]\to [c.d]](https://s0.wp.com/latex.php?latex=p%3A%5Ba%2Cb%5D%5Cto+%5Bc.d%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
of the form p(x)=mx+k that maps a to c and b to d. This is called the positive linear map of [a,b] to [c,d] because its graph is a straight line with positive slope.
in X as follows: On [0,a] it equals the positive linear map of [0,a] to I=[0,1] followed by f; on [a,b] it equals the positive linear map of [a,b] to I followed by g; on [b,1] it equals the positive linear map of [b,1] to I followed by h. This path 
is path homotopic to 
be the map whose graph is pictured in Figure 51.9 (taken from Munkre’s Book)
. Let’s see why this is so.
is the positive linear map of [0,a] to [0,c], followed by the positive linear map of [0,c] to I, followed by f. This equals the positive linear map of [0,a] to I, followed by f, which is precisely 
is a path in I from 0 to 1, and so is the identity map 
. Since I is convex, there is a path homotopy P in I between p and i. Then, 
is a path homotopy in X between 
.
is exactly the triple product 
and 
.![(g*h)(s)=\begin{cases} g(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ h(2s-1)\ &\text{for }s\in [\frac{1}{2},1] \end{cases}](https://s0.wp.com/latex.php?latex=%28g%2Ah%29%28s%29%3D%5Cbegin%7Bcases%7D++++g%282s%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%5C%5C++++h%282s-1%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C1%5D++++%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![f*(g*h)(s)=\begin{cases} f(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ (g*h)(2s-1)\ &\text{for }s\in [\frac{1}{2},1] \end{cases} =\begin{cases} f(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ g(4s-2)\ &\text{for }s\in [\frac{1}{2},\frac{3}{4}]\\ h(4s-3) &\text{for }s\in [\frac{3}{4},1] \end{cases}](https://s0.wp.com/latex.php?latex=f%2A%28g%2Ah%29%28s%29%3D%5Cbegin%7Bcases%7D++++f%282s%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%5C%5C++++%28g%2Ah%29%282s-1%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C1%5D++++%5Cend%7Bcases%7D++++%3D%5Cbegin%7Bcases%7D++++f%282s%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%5C%5C++++g%284s-2%29%5C+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C%5Cfrac%7B3%7D%7B4%7D%5D%5C%5C++++h%284s-3%29+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B%5Cfrac%7B3%7D%7B4%7D%2C1%5D++++%5Cend%7Bcases%7D++++&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
when c=1/4 and d=1/2. Thus, the these two products are path homotopic, and we have finally proven the associativity of *.
, where 
is the threshold of hearing, which is the smallest sound a human can hear.
-algebra X.
) is an increasing sequence in X, then
) is a decreasing sequence in X and if 
, then
. A decreasing sequence means the opposite, i.e. 
.
for some n, then both sides of the equation are 
, and inequality holds. Henceforth, we can just consider the case 
for all n.
and 
for n>1. Then 
is a disjoint sequence of sets in X such that
, 
(since (
) is a disjoint sequence of sets)
, we have that 
for n>1, so the finite series on the right side telescopes to become
, so that 
is an increasing sequence of sets in X.![\begin{aligned} \mu (\bigcup_{n=1}^\infty E_n) &=\lim \mu (E_n)\\ &=\lim [\mu (F_1)-\mu (F_n)]\\ &=\mu (F_1) -\lim \mu (F_n) \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D++++%5Cmu+%28%5Cbigcup_%7Bn%3D1%7D%5E%5Cinfty+E_n%29+%26%3D%5Clim+%5Cmu+%28E_n%29%5C%5C++++%26%3D%5Clim+%5B%5Cmu+%28F_1%29-%5Cmu+%28F_n%29%5D%5C%5C++++%26%3D%5Cmu+%28F_1%29+-%5Clim+%5Cmu+%28F_n%29++++%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
, it follows that
.
, let 
denote the constant path 
mpping all of I to the point x. If f is a path in X from 
to 
, then ![[f]*[e_{x_1}]=[f]](https://s0.wp.com/latex.php?latex=%5Bf%5D%2A%5Be_%7Bx_1%7D%5D%3D%5Bf%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![[e_{x_0}]*[f]=[f]](https://s0.wp.com/latex.php?latex=%5Be_%7Bx_0%7D%5D%2A%5Bf%5D%3D%5Bf%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
be the path defined by 
. ![[f]*[\bar{f}]=[e_{x_0}]](https://s0.wp.com/latex.php?latex=%5Bf%5D%2A%5B%5Cbar%7Bf%7D%5D%3D%5Be_%7Bx_0%7D%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
and ![[\bar{f}]*[f]=[e_{x_1}]](https://s0.wp.com/latex.php?latex=%5B%5Cbar%7Bf%7D%5D%2A%5Bf%5D%3D%5Be_%7Bx_1%7D%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
is a continuous map, and if F is a path homotopy in X between the paths f and f’, then 
is a path homotopy in Y between the paths 
and 
.
is a continuous map and if f and g are paths in X with f(1)=g(0), then
, where h=f*g as defined previously.
.
denote the constant path in I at 0, and we let 
is also a path in I from 0 to 1.
(Straight-line homotopy) Then 
is a path homotopy in X between the paths 
and 
(Lemma 1). Furthermore by Lemma 2, 
which is equivalent to 
.
denotes the constant path at 1, then 
is path homotopic in I to the path i, shows that 
. Then 
is a path in I beginning and ending at 0. The constant path 
is also beginning and ending at 0. Again, because I is convex, there is a path homotopy H in I between 
is path homotopy between 
and 
. Very similarly, we can use the fact that 
is path homotopic in I to 
. It covers the areas of Algebra, Geometry, Counting and Probability, and Number Sense, over 500 examples and problems with fully explained solutions.
and measures should be additive over disjoint sets in X.
for all 
) of sets in X, then
.
, we say it is finite. More generally, if there exists a sequence 
and such that 
for all n, then we say that 
be definied on X by 
, for all 
. We can see that 
be defined by 
, 
if 
. 
defined on B which coincides with length on open intervals. I.e. if E is the nonempty interval (a,b), then 
. This measure is usually called Lebesgue measure (or sometimes Borel measure). It is not a finite measure since 
. But it is 
, we define the product f*g of f and g to be he path h given by the equations![h(s)=\begin{cases}f(2s) &\text{for }s\in [0,\frac{1}{2}], \\ g(2s-1)& \text{for }s\in[\frac{1}{2}, 1]\end{cases}](https://s0.wp.com/latex.php?latex=h%28s%29%3D%5Cbegin%7Bcases%7Df%282s%29+%26%5Ctext%7Bfor+%7Ds%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%2C+%5C%5C+g%282s-1%29%26+%5Ctext%7Bfor+%7Ds%5Cin%5B%5Cfrac%7B1%7D%7B2%7D%2C+1%5D%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![[f]*[g]=[f*g]](https://s0.wp.com/latex.php?latex=%5Bf%5D%2A%5Bg%5D%3D%5Bf%2Ag%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![H(s,t)=\begin{cases}F(2s,t) &\text{for }s\in[0,\frac{1}{2}],\\ G(2s-1,t)&\text{for }s\in[\frac{1}{2},1]\end{cases}](https://s0.wp.com/latex.php?latex=H%28s%2Ct%29%3D%5Cbegin%7Bcases%7DF%282s%2Ct%29+%26%5Ctext%7Bfor+%7Ds%5Cin%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%2C%5C%5C+G%282s-1%2Ct%29%26%5Ctext%7Bfor+%7Ds%5Cin%5B%5Cfrac%7B1%7D%7B2%7D%2C1%5D%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
.
belongs to X for every set E belonging to Y.
the set 
belongs to X.
for every Borel set E.
.
if f and f’ are path homotopic.
and 
are equivalence relations.
. The map F(x,t) is the required homotopy.
. Let F be a homotopy between f and f’. We can then verify that G(x,t) = F(x, 1-t) is a homotopy between f’ and f.
, we show that 
. Let F be a homotopy between f and f’, and let F’ be a homotopy between f’ and f”. This time, we need to define a slightly more complicated homotopy G: X x I -> Y by the equation![G(x,t) = \begin{cases} F(x,2t) &\text{for }t\in [0,\frac{1}{2}],\\ F'(x, 2t-1) &\text{for } t\in [\frac{1}{2}, 1].\end{cases}](https://s0.wp.com/latex.php?latex=G%28x%2Ct%29+%3D+%5Cbegin%7Bcases%7D+F%28x%2C2t%29+%26%5Ctext%7Bfor+%7Dt%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%2C%5C%5C+F%27%28x%2C+2t-1%29+%26%5Ctext%7Bfor+%7D+t%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C+1%5D.%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
