Are non-computerized mathematical "recreations" dead?
Are non-computerized mathematical "recreations" dead?
Sandbox for my rant, but I would be interested in hearing opinions, particularly from those who never knew a world where programmable computers were not ubiquitous and cheap.
First, to be clear, I think there will always be an interest in "puzzle math", competitive problem solving of the sort seen in Math Olympiad, and competitive solving of popular puzzles like Rubik's Cube. There will also be serious mathematicians who may have little use for computers except as word processors. I'm talking about something a little different here. Since it's hard to define, I'll give an instance: flexagons, which I would explain as a method by which a strip of paper can be folded so it is isomorphic to an oriented binary tree and a traversal can be carried out by a series of folding and unfolding steps. I first encountered them in Madachy's Mathematical Recreations (hence my subject line) as a teen and I forget the other ones in that book. I guess magic squares would qualify as a mathematical recreation, but I have always been more excited by those that involve some craft: cutting, folding, gluing, etc.
Of course, since I did have access to a microcomputer at the time (though it was less common) one of the things I did a few years later was to code up the algorithm from the book on a computer (which is optimized for a human writing out a paper table, as I eventually realized).
So, examples aside, I have kids in their teens, and throughout their childhood I have shown them some "craft" mathematics that I find interesting (admittedly often using a computer as a time-saver, but it's not required). Here, for instance is an icosahedron and dodecahedron colored to illustrate how their rotation groups are equivalent to the group of 60 even permutations of five elements (pardon the sloppy glue work). Anyway, they were underwhelmed, as with most such examples. "Cool thingy, dad! Now let me get back to my computer."
It may be an overall lack of interest, since this kind of activity has always had limited appeal, but I find that I am increasingly fascinated by the process of pulling mathematics away from electronic computers where everything is just way too easy, and constructing physical analogs. My kids think this is just weird, and they may be right.
In my estimation, we ought to be living in a "golden age of non-computerized mathematical recreations" since computers themselves can be used to discover and optimize the kinds of things that can be built with crafts (e.g. use a 3D printer to make prototypes until you find one that is easy to craft by hand). There are also newer craft products like polymer clay and a lot of interest in crafts in general (stores like Michaels), though rarely aimed at mathematics. The field looks wide open.
So what I wonder is if the main historical appeal of these crafts, such as flexagons, is that they could be used as analog devices before computers existed. In this view, they are now as obsolete as slide rules, given the existence of computers--in which not only can a binary tree be constructed, but a flexagon itself can be modeled and rendered in animation.
The question is whether it's interesting to have touchable mathematics. I mean, aside from manipulatives in a Montessori preschool. Are these things of any interest to teens and adults with access to powerful electronic computers?
First, to be clear, I think there will always be an interest in "puzzle math", competitive problem solving of the sort seen in Math Olympiad, and competitive solving of popular puzzles like Rubik's Cube. There will also be serious mathematicians who may have little use for computers except as word processors. I'm talking about something a little different here. Since it's hard to define, I'll give an instance: flexagons, which I would explain as a method by which a strip of paper can be folded so it is isomorphic to an oriented binary tree and a traversal can be carried out by a series of folding and unfolding steps. I first encountered them in Madachy's Mathematical Recreations (hence my subject line) as a teen and I forget the other ones in that book. I guess magic squares would qualify as a mathematical recreation, but I have always been more excited by those that involve some craft: cutting, folding, gluing, etc.
Of course, since I did have access to a microcomputer at the time (though it was less common) one of the things I did a few years later was to code up the algorithm from the book on a computer (which is optimized for a human writing out a paper table, as I eventually realized).
So, examples aside, I have kids in their teens, and throughout their childhood I have shown them some "craft" mathematics that I find interesting (admittedly often using a computer as a time-saver, but it's not required). Here, for instance is an icosahedron and dodecahedron colored to illustrate how their rotation groups are equivalent to the group of 60 even permutations of five elements (pardon the sloppy glue work). Anyway, they were underwhelmed, as with most such examples. "Cool thingy, dad! Now let me get back to my computer."
It may be an overall lack of interest, since this kind of activity has always had limited appeal, but I find that I am increasingly fascinated by the process of pulling mathematics away from electronic computers where everything is just way too easy, and constructing physical analogs. My kids think this is just weird, and they may be right.
In my estimation, we ought to be living in a "golden age of non-computerized mathematical recreations" since computers themselves can be used to discover and optimize the kinds of things that can be built with crafts (e.g. use a 3D printer to make prototypes until you find one that is easy to craft by hand). There are also newer craft products like polymer clay and a lot of interest in crafts in general (stores like Michaels), though rarely aimed at mathematics. The field looks wide open.
So what I wonder is if the main historical appeal of these crafts, such as flexagons, is that they could be used as analog devices before computers existed. In this view, they are now as obsolete as slide rules, given the existence of computers--in which not only can a binary tree be constructed, but a flexagon itself can be modeled and rendered in animation.
The question is whether it's interesting to have touchable mathematics. I mean, aside from manipulatives in a Montessori preschool. Are these things of any interest to teens and adults with access to powerful electronic computers?
Re: Are non-computerized mathematical "recreations" dead?
Well, computers are really the only things nowadays.
A 17 year old guy that have useless discoveries in life and other rules.
My actions weren't smart back then.
Code: Select all
x = 13, y = 20, rule = B3/S23
11b2o$11b2o4$8b2o$8b2o2$2o$2o3$3b2o$3b2o2$11b2o$10b2o$12bo$3b2o$3b2o!
Re: Are non-computerized mathematical "recreations" dead?
You've persuaded me that if my CA tiles are ever going to take off, I probably need a phone app to go with them. This might even be fun (I am not a great judge of this). After placing the physical tiles, you should be able to snap a picture and have the whole layout scanned and imported into your phone. This is easier than most image recognition problems and shouldn't be too hard to code. Of course, if you have your phone, you don't need the tiles... I still like manipulating physical objects.
-
- Posts: 2200
- Joined: August 5th, 2016, 10:27 am
- Location: 拆哪!I repeat, CHINA! (a.k.a. 种花家)
- Contact:
Re: Are non-computerized mathematical "recreations" dead?
I'm sure it is, for me, who is fond of chemical structures and therefore the underlying mathematics. For instance, sometime ago I was really longing for a flexible physical model of Goldberg polyhedra, visualizing how the icosahedral symmetry groups are applied to fullerene geometry. But it turns out that magnetic ball-and-stick models can't construct stable pentagons/hexagons with variable sides, while Zometool, a model that is good at handling icosahedral symmetry(no ad intended), doesn't support the rotational group.pcallahan wrote:The question is whether it's interesting to have touchable mathematics.
Another stuff I'm concerned about are the space-filling polyhedrons, for a better understanding about crystallography.
熠熠种花 - Glimmering Garden
Harvest Moon
2-engine p45 gliderless HWSS gun
Small p2070 glider gun
Forgive me if I withhold my enthusiasm.
Harvest Moon
2-engine p45 gliderless HWSS gun
Small p2070 glider gun
Forgive me if I withhold my enthusiasm.
Re: Are non-computerized mathematical "recreations" dead?
I'd say I agree with that. It is interesting enough to have such things demonstrating chemistry. It is better than something like MolView MooseyView. I don't think they are serious enough to be researched though. Though, LeapLife isnt serious too but it could be investigated...GUYTU6J wrote: ↑February 26th, 2020, 3:37 amI'm sure it is, for me, who is fond of chemical structures and therefore the underlying mathematics. For instance, sometime ago I was really longing for a flexible physical model of Goldberg polyhedra, visualizing how the icosahedral symmetry groups are applied to fullerene geometry. But it turns out that magnetic ball-and-stick models can't construct stable pentagons/hexagons with variable sides, while Zometool, a model that is good at handling icosahedral symmetry(no ad intended), doesn't support the rotational group.pcallahan wrote:The question is whether it's interesting to have touchable mathematics.
Another stuff I'm concerned about are the space-filling polyhedrons, for a better understanding about crystallography.
-
- Posts: 2200
- Joined: August 5th, 2016, 10:27 am
- Location: 拆哪!I repeat, CHINA! (a.k.a. 种花家)
- Contact:
Re: Are non-computerized mathematical "recreations" dead?
Tried to solve the nonisorulemin-S012345678 still life problem with 256 hand-made squares. Largest C4_1 partial:
This method doesn't seem effective, but making a 256-state ruletable in golly with markings like these and drawing isn't easy either.熠熠种花 - Glimmering Garden
Harvest Moon
2-engine p45 gliderless HWSS gun
Small p2070 glider gun
Forgive me if I withhold my enthusiasm.
Harvest Moon
2-engine p45 gliderless HWSS gun
Small p2070 glider gun
Forgive me if I withhold my enthusiasm.
Re: Are non-computerized mathematical "recreations" dead?
I actually made a ruletable for things like this: https://conwaylife.com/forums/viewtopic ... 780#p79780
It should easily modifiable to do other rules, or rather the script used to make the ruletable should be modifiable. Here's the script
Code: Select all
# Born as state 1
# Turn into wire
# Turn into state 2
# Regular life transitions but to state 1
iconHeader = '''
XPM
"7 1743 2 1"
". c #303030"
"O c #FFFFFF"
'''
RULE = ['@RULE NotWireLife',
'@TABLE',
'n_states:251',
'neighborhood:Moore',
'symmetries:none']
def applyPadding(s):
return '0'*(8-len(s))+s
def applyPadding2(s):
return s+'0'*(8-len(s))
def writeVars():
var = []
states = list(map(str,range(1,251)))
var.append('var a={'+'0,'+','.join(states)+'}')
var.append('var b={'+'0,'+','.join(states)+'}')
var.append('var c={'+'0,'+','.join(states)+'}')
var.append('var d={'+'0,'+','.join(states)+'}')
var.append('var e={'+'0,'+','.join(states)+'}')
var.append('var f={'+'0,'+','.join(states)+'}')
var.append('var g={'+'0,'+','.join(states)+'}')
var.append('var h={'+'0,'+','.join(states)+'}')
return var
def writeLifeTransitions():
trans = []
n = 0
t = ''
o = ''
#birth
for b in range(256):
n = bin(b)[2:].replace('1','2')
t = '0'+applyPadding(n)
if n.count('2')==3:
o = '1'
trans.append(','.join(list(t+o)))
#survival
for s in range(256):
n = bin(s)[2:].replace('1','2')
t = '2'+applyPadding(n)
if not(n.count('2')==2 or n.count('2')==3):
o = '0'
trans.append(','.join(list(t+o)))
return trans
def writeWireToStateTwoTransitions():
t = []
for i in range(3,250):
t.append(str(i)+',a,b,c,d,e,f,g,h,'+'2')
return t
def writeWireTransitions():
trans = ['1,0,0,0,0,0,0,0,0,3']
n = 0
t = ''
o = ''
i = 4
special = {'1':'15',
'01':'31',
'001':'64',
'0001':'131',
'00001':'15',
'000001':'31',
'0000001':'64',
'00000001':'131'}
for b in range(1,256):
n = bin(b)[2:][::-1]
t = ','.join(list('1'+applyPadding2(n)))
o = ','+str(i)
if n.count('1')!=1:
trans.append(t+o)
i+=1
else:
trans.append(t+','+special[n])
return trans
def writeIcons():
template ='''
"uunnnmm"
"uu.n.mm"
"w.unm.e"
"wwwOeee"
"w.fsq.e"
"ff.s.qq"
"ffsssqq"
'''
#u = nw
#m = ne
#f = sw
#q = set
icons = [
'''
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
''',
'''
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
"OOOOOOO"
''',
'''
"......."
"......."
"......."
"...O..."
"......."
"......."
"......."
''']
n = 0
temp = ''
for b in range(3,256):
temp = '''
"uunnnmm"
"uu.n.mm"
"w.unm.e"
"wwwOeee"
"w.fsq.e"
"ff.s.qq"
"ffsssqq"
'''
n = bin(b)[2:][::-1]
n = applyPadding2(n)
if n[0]=='1':
temp = temp.replace('n','O')
if n[1]=='1':
temp = temp.replace('m','O')
if n[2]=='1':
temp = temp.replace('e','O')
if n[3]=='1':
temp = temp.replace('q','O')
if n[4]=='1':
temp = temp.replace('s','O')
if n[5]=='1':
temp = temp.replace('f','O')
if n[6]=='1':
temp = temp.replace('w','O')
if n[7]=='1':
temp = temp.replace('u','O')
temp = temp.replace('n','.')
temp = temp.replace('m','.')
temp = temp.replace('e','.')
temp = temp.replace('q','.')
temp = temp.replace('s','.')
temp = temp.replace('f','.')
temp = temp.replace('w','.')
temp = temp.replace('u','.')
if n.count('1')!=1:
icons.append(temp)
return icons
def writeColors():
c = []
for i in range(1,251):
c.append(str(i)+' 255 255 255')
return c
res = []
res.extend(RULE)
res.extend(writeVars())
res.extend(writeWireTransitions())
res.extend(writeWireToStateTwoTransitions())
res.extend(writeLifeTransitions())
res.append('2,a,b,c,d,e,f,g,h,1')
res.append('@ICONS')
res.append(iconHeader)
res.extend(writeIcons())
res.append('@COLORS')
res.extend(writeColors())
file = open('notwirelife.txt','a')
file.write('\n'.join(res))
file.close()
Code: Select all
x = 17, y = 33, rule = WireLife
6.6A.2A$6.6A.2A$13.2A$6.2A5.2A$6.2A5.2A$6.2A5.2A$6.2A$6.2A.6A$6.2A.6A
5$.A$A$3A9.A2.A$16.A$12.A3.A$13.4A3$13.A$11.A3.A$16.A$11.A4.A$12.5A3$
12.2A$10.A4.A$16.A$10.A5.A$11.6A!
Re: Are non-computerized mathematical "recreations" dead?
What is this problem about? I presume this is a problem about finding a 256-cell pattern where every cell has a unique neighbourhood configuration.
Still drifting.
-
- Posts: 2200
- Joined: August 5th, 2016, 10:27 am
- Location: 拆哪!I repeat, CHINA! (a.k.a. 种花家)
- Contact:
Re: Are non-computerized mathematical "recreations" dead?
Yes, the orginal discussion is in the endemic still lives thread. If you have solutions make sure to post there!
Good job, but here the problem is unrelated to GoL and no "detect" state is needed. I guess it can be trivially modified to contain the full 256 transitions and kill the cells with actual neighbours that are inconsistent with assigned neighbours.
熠熠种花 - Glimmering Garden
Harvest Moon
2-engine p45 gliderless HWSS gun
Small p2070 glider gun
Forgive me if I withhold my enthusiasm.
Harvest Moon
2-engine p45 gliderless HWSS gun
Small p2070 glider gun
Forgive me if I withhold my enthusiasm.
Re: Are non-computerized mathematical "recreations" dead?
Yeah, I'm aware of both points, but the detect state is handy so you dont have to search for the right state to use in golly.