un·pre·ten·tious·ness

Computing Nim

2010-11-11T06:21:00.003-07:00

A few Math Club meetings ago we learned about the combinatorial game called Nim. I decided to play around with Nim in Haskell and came up with this bit of literate Haskell which you can paste into a text file with the extension ".lhs" and run with your favorite haskell compiler (GHC of course).

The game is a simple two player game. Start with some piles with various amounts of objects. Each player in turn takes at least one object from a single pile. The winning player is the one that takes the last object.

If you are coming from a math background you might want to keep this reference open.

For our Haskell code we start with some standard imports.

> module GameNormal where
> 
> import System.IO
> import Data.Maybe
> import Data.Bits
> import Data.List

We declare a data type to abstract out the idea of a player:

> data Player = Player { name :: String, play :: [Int] -> IO (Int,Int) }

This means that a player has a name and a strategy for playing in the form of a function from a list of piles to a pair where the first entry is the index of the pile to take from and the second entry is the amount to take (we will ignore the IO for now other than to say that it will allow us to make a play strategy that interacts with the user).

What would the simplest play strategy look like?

> naivePlay :: [Int] -> (Int,Int)
> naivePlay _ = (0,1)

This states that we ignore the input and always choose one object from the first pile (0 is our first index of course). Once we know what our play will be we need to produce a new pile. We will call this function draw and it will have the type:

> draw :: Int -> Int -> [Int] -> [Int]

The first base case is when we have no piles to begin with. This condition should never happen so we define it to be an error.

> draw _ _ [] = error "Invalid pile."

The next base case is when we are drawing from the first pile. In this case we construct a new list of piles where the first element is the original first element p minus the number to draw n.

> draw 0 n (p:ps) = (p - n) : ps

Finally for other indexes we construct a new list with the existing first element and call draw on the rest of the list with the index reduced by one.

> draw i n (p:ps) = p : draw (i-1) n ps

At this point we can test what we have so far in ghci.

*Main> naivePlay [4,3,2]
(0,1)
*Main> draw 0 1 [4,3,2]
[3,3,2]
*Main> draw 2 2 [3,3,2]
[3,3,0]

We can see that draw is producing a new list of pile as expected. If all the elements in a pile are taken we end up with a pile with zero elements. We can remove the empty piles by filtering them out:

*Main> :t filter
filter :: (a -> Bool) -> [a] -> [a]
*Main> filter (> 0) [3,3,0]
[3,3]

These two functions can of course be composed:

*Main> :t (filter (> 0) . draw 2 2)
(filter (> 0) . draw 2 2) :: [Int] -> [Int]
*Main> (filter (> 0) . draw 2 2) [3,3,2]
[3,3]

Now we turn to winning strategy. We learned in Math Club to look at the piles in their binary representation and if at the end of a player's turn they can end up with an even number of bits for each column in that representation then that player will win. For example [4,3,2] can be written:

100
 11
 10

We can compute the evenness of the columns with the xor (exclusive or) logic operation:

100
 11
 10
--- xor
101

Any columns with 1's after the computation mean that that column has an odd number of 1's. The problem now is how do we pick a single pile to change so that the xor computation results in 0. One option is to just try every possibility until we find one that works. We can calculate the xor of a list of piles with a fold as in the following function:

> xorAll :: [Int] -> Int
> xorAll = foldl1 xor

We can think of a fold like we think of the mathematical sum but with the operation as an argument to the fold function. Folds are also specified to be either right (foldr) or left (foldl) associative. In Haskell the variants ending in a 1 mean that they assume the list has at least one element and use that as the starting value for the running total. So if we use the symbol for xor then xorAll is similar to . Running we get:

*Main> xorAll [4,3,2]
5

We will also need to be able to generate all possible plays given a list of piles:

> allPlays :: [Int] -> [(Int,Int)]
> allPlays ps = [(i,n) | i <- [0..(length ps -1)], n <- [1..(ps !! i)]]

This code uses a list comprehension which looks very similar to what can be written in typical mathematical notation: where is the number of piles and is the amount in pile . Now we have all we need to write the strategy:

brutePlay :: [Int] -> (Int,Int)
brutePlay ps = (head . filter isWinningPlay . allPlays) ps
  where isWinningPlay (i,n) = xorAll (draw i n ps) == 0

We could say that brutePlay is defined to be the first (head) winning (filter isWinningPlay) play (allPlays), but what if there isn't a winning play?

*Main> brutePlay [2,2]
*** Exception: Prelude.head: empty list

It isn't very sportsmanlike for a program to throw an exception just because it doesn't have a winning play, so we amend:

> brutePlay :: [Int] -> (Int,Int)
> brutePlay ps = fromMaybe (0,1) winningPlays
>   where isWinningPlay (i,n) = xorAll (draw i n ps) == 0
>         winningPlays = (listToMaybe . filter isWinningPlay . allPlays) ps

Lets try this out on the examples we had before:

*Main> brutePlay [2,2]
(0,1)
*Main> brutePlay [4,3,2]
(0,3)

Pulling it all together we can now run a game by alternating turns and drawing on the current pile until we hit the base case of having no piles left.

> runGame :: Player -> Player -> [Int] -> IO ()
> runGame a b [] = putStrLn ("The winner is: " ++ name b)
> runGame a b p  = do
>     putStrLn . concat $ [name a, "'s turn.  Piles are: ", show p]
>     (i,n) <- play a p
>     runGame b a . filter (>0) . draw i n $ p 
> 
> main :: IO ()
> main = runGame a b ps
>   where a = Player "Brute" (return . brutePlay)
>         b = Player "Naive" (return . naivePlay)
>         ps = [4,3,2]

Loading in ghci we can run and see the outcome:

*Main> :main
Brute's turn.  Piles are: [4,3,2]
Naive's turn.  Piles are: [1,3,2]
Brute's turn.  Piles are: [3,2]
Naive's turn.  Piles are: [2,2]
Brute's turn.  Piles are: [1,2]
Naive's turn.  Piles are: [1,1]
Brute's turn.  Piles are: [1]
The winner is: Brute
*Main>

We can do better though. If we look at allPlays we will notice that it returns a list with a length equal to the sum of the entries. Instead let's consider what value the xor of the piles would be if we took away an arbitrary pile we will call i. Lets assume there were a objects in the pile we are removing. If the resulting xor of the remaining piles it is zero then we know what move we should make (take a objects from i). If it is some non-zero value b then if we put back into pile i b objects then we know that the xor of the new piles would be zero (using the identity that any value xor itself is zero). As long as b is smaller than a then we will be able to do this. This means that now we only consider one possible play for each pile.

> smartPlay :: [Int] -> (Int,Int)
> smartPlay ps = fromMaybe (0,1) winningPlays
>   where s  = xorAll ps
>         bs = map (xor s) ps
>         ds = zipWith (-) ps bs
>         ts = zip [0..] ds
>         winningPlays = (listToMaybe . filter ((> 0) . snd)) ts

Breaking this down:

s is the xor of all the piles.
bs is the xor of all the piles but the pile at index i of `bs.
ds is the element-wise difference between the original pile and bs.
ts is the pairing of indexes with the differences.
winningPlays is Nothing or the first element in ts with a positive difference.

Finally the following code allows interactive play. Notice that we need the IO type in promptPlay because we are interacting with the environment.

> promptPlay :: [Int] -> IO (Int,Int)
> promptPlay p = do
>     j <- promptInt "Enter pile to draw from" 1 (length p)
>     let i = j - 1
>     n <- promptDraw (p !! i)
>     return (i,n)
> 
> promptDraw :: Int -> IO Int
> promptDraw 1 = putStrLn "There is only one object.  You take it." >> return 1
> promptDraw n = do
>     putStrLn . concat $ ["That pile has ", show n, " objects."]
>     promptInt "Enter number to draw" 1 n
> 
> parseInt :: String -> Maybe Int
> parseInt = listToMaybe . map fst . reads
> 
> inRange :: Int -> Int -> Int -> Maybe Int
> inRange a b n = if a <= n && n <= b then Just n else Nothing
> 
> promptInt :: String -> Int -> Int -> IO Int
> promptInt p a b = do
>     putStr . concat $ [p, " (", show a, ", ", show b, "): "]
>     r <- getLine
>     maybe (promptInt p a b) return (parseInt r >>= inRange a b)

(Created with pandoc: pandoc -f markdown+lhs -t html+lhs .\input.lhs -o output.html --offline --webtex)

Haskell for Math People

2010-11-11T06:12:00.003-07:00

I'd like to keep a little quick reference here of tips for reading Haskell code if you are coming from a math background:

Function application is left associative and doesn't use parenthesis so f a b c is the same as (((f a) b) c) which is the same as the normal math notation .
Writing function application this way makes it easy to do partial application. For instance if we have a math function we can think of it as function . That is a function from a space to a space of functions. This function would naturally take one parameter and the "value" that it represents would also take one parameter. To partially apply in Haskell we just provide the number of parameters we want to apply.
We often talk about "types" in Haskell and this is somewhat analogous to making statements about the domain and range of a function. For example when we write f :: Int -> Int -> Int in Haskell we are saying that we have a function named where . More precicely types are isomorphic to proofs in constructive mathmatics (see Curry–Howard correspondence).
The following program often uses a data structure called a list. We write the type of a list as [a] where a is some type. We construct lists in a few ways
1. As a list of values: [1,2,3].
2. Using the : (called "Cons") operator: 1:2:3:[]. The Cons operator is a function that takes a value and a list and produces a new list with that value at the head of the list. The empty list is written as [].
3. As a list comprehension which we will talk about later.
Function definitions in haskell are written with the function name followed by pattern matched arguments, an equals sign, then an expression in terms of the arguments given. We call everything to the left of the equals sign the "left-hand side" or lhs and everything to the right of the equals the "right-hand side" or rhs. Pattern matching is where instead of listing a name for an argument we specify some structure or pattern that the value must fit for the function to be applicable. Haskell will look for the first applicable function and use that expression. For example the function f (x:xs) = (1 + x) : f xs is applicable when given a list. The first element of the list will be bound to x on the rhs while the rest of the list (the tail) will be bound to xs on the rhs. Attempting to apply f to [] would result in a "Non-exhaustive patterns" exception because [] is the empty list and does not have a head and a tail. To fix this we would have to define that base case with f [] = []. We now have a function where given a list of numbers it would result in a new list of numbers where each number is incremented by one.
Function composition is written with a period f . g which is meant to mimic the math notation .

Slang.

2010-06-04T03:44:00.004-06:00

From John Badcock's:

Slang.
A Dictionary
of
the Turf, the Ring, the Chase, the Pit, of Bon-Ton,
and the
Varieties of Life,
Forming the Completest and Most Authentic
Lexicon Balatronicum
Hitherto Offered to the Notice of
the Sporting World,
For elucidating Words and Phrases that are necessarily, or purposely,
cramp, mutative, and unintelligible, outside their
respective Spheres.
Interspersed with
Anecdotes and Whimsies,
with Tart Quotations, and Rum-Ones;
with Examples, Proofs, and Monitory Precepts,
Useful and Proper for
Novices, Flats, and Yokels.

Computational Thinking

2010-05-02T08:18:00.002-06:00

Alan Jacobs linked to this article recently (PDF):

http://www.cs.cmu.edu/afs/cs/usr/wing/www/publications/Wing06.pdf

I agree with much of what Wing says. I think the article is motivated by a general trend in Computer Science higher education that amounts to CS departments failing to "attract" good students. On the other hand it remains true that computational thinking applied to many other fields (essentially science in general) continues to be the foremost innovation for that field. Wing gets the distinction right that it isn't about programming. The example I always use is that cellular biology must be concerned with how cells communicate and act on each other. This problem has it's theoretical underpinnings in computer science. Not on using computers to run simulations on how cells might communicate (while that can be an interesting thing to study), but on answering questions like "What ways can discrete objects communicate?"

So why, given the importance of computational thinking to so much of science, is Computer Science failing to attract good students who want "go on to a career in medicine, law, business, politics, any type of science or engineering, and even the arts."? In my experience the public perception of CS is that it is all about programming or all about computer hardware. Computer Scientists most important body part is their fingers because that works the keyboard right? The cultural stereotype that gets assigned to Computer Scientists is the geek or nerd. As a consequence people who identify with that stereotype (for whatever reason) see themselves as cast in the role of Computer Scientist regardless of their ability or desire to reason well computationally. The ETS's major field test reports (pdf) that only 28% of CS seniors have an overall GPA above 3.5. Only 36% have that high a GPA in their major. Mathematics comes in at 47% and 43%.

Somehow Mathematics is able to attract thinking students while still getting a similar stereotyping from broader culture and I can't help but think that has to do with the seat it has a the education table as a first class member (never mind that the system at large fails to emphasize the important parts of mathematics). I like Wing's vision of a world where Computational Thinking has a place at that table (though I shudder to think of the tortures the education system could inflict in CS's name).

The man of action

2010-04-26T05:40:00.003-06:00

To the Greeks the Private Realm was the sphere of life ruled by the necessity of sustaining life, and the Public Realm the sphere of freedom where a man could disclose himself to others. Today, the significance of the terms private and public has been reversed; public life is the necessary impersonal life, the place where a man fulfills his social function, and it is in his private life that he is free to be his personal self.

In consequence the arts, literature in particular, have lost their traditional principal human subject, the man of action, the doer of public deeds.
—W. H. Auden

More LINQ translation.

2010-04-20T08:46:00.004-06:00

We will get to the second part of the last post later, but right now I wanted to post another quick "convert to Haskell from LINQ" entry this time from a post by Eric Lippert.


> import Control.Monad

Notice how similar this definition is to the C# version:


> data Tree = Node Tree Tree | Empty

In both version Node is the name of the constructor taking two arguments. Both cases make an immutable "object".


> instance Show Tree where
>   show Empty = "x"
>   show (Node l r) = "(" ++ show l ++ show r ++ ")"

The C# version's BinaryTreeString has a helper function doing the recursion, but it isn't really necessary (in the C# or Haskell). What the C# gains is the ability to side-effect on a StringBuilder directly rather than returning partial results.


> allTrees :: Int -> [Tree]
> allTrees 0 = [Empty]
> allTrees n = [Node l r | i <- [0..n-1]
>                        , l <- allTrees i
>                        , r <- allTrees (n - 1 - i)
>                        ]

This translates very directly from the LINQ code, each from statement becomes a statement with a <-. That's it! Testing we have:


> trees = mapM_ (putStrLn . show) (allTrees 4)

giving the output:


*Main> trees
(x(x(x(xx))))
(x(x((xx)x)))
(x((xx)(xx)))
(x((x(xx))x))
(x(((xx)x)x))
((xx)(x(xx)))
((xx)((xx)x))
((x(xx))(xx))
(((xx)x)(xx))
((x(x(xx)))x)
((x((xx)x))x)
(((xx)(xx))x)
(((x(xx))x)x)
((((xx)x)x)x)

In the comments we have several answers to the challenge the last of which translates to this:


> data ATree = ANode [ATree]

> instance Show ATree where
>   show (ANode [])   = "{}"
>   show (ANode xs) = "{" ++ (foldl1 (++) (map show xs)) ++ "}"

> allATrees :: Int -> [ATree]
> allATrees n = [ANode x | x <- h (n - 1)]
>   where h 0 = [[]]
>         h n = [(ANode l) : r | i <- [1..n]
>                              , l <- h (i - 1)
>                              , r <- h (n - i)
>                              ]

> aTrees = mapM_ (putStrLn . show) (allATrees 4)

Evaluating aTrees gives:


*Main> aTrees
{{}{}{}}
{{}{{}}}
{{{}}{}}
{{{}{}}}
{{{{}}}}

Note that this output is all strings of nested braces that are in a single outer pair of braces and are eight characters long. If we called allATrees's helper function h directly we would be removing the "single outer pair of braces" constraint. So h of n is of the same order as allTrees.

Solving Combinatory Problems with Haskell

2010-04-15T07:08:00.006-06:00

Today my Visual Studio start page had a link to an interesting blog post by Octavio Hernandez and I couldn't pass up the opportunity to solve the problem with Haskell. Please note that this is not in anyway a commentary on language X is better than Language Y, rather looking at the same problem in many different languages is very helpful in understanding the problem itself better. I'll try and make this a literate Haskell post so you should be able to copy and paste the text into a text file with the ".lhs" extension and have everything run. So we begin by importing an operator that we will use later:


> import Data.List ((\\))

We will need a list of the digits.


> num = [1..9]

The first generation is all of the one digit numbers that satisfy n mod 1 == 0. That's an easy one, all of them satisfy that condition!


> m1 = num

We can see this in the C# version as from i1 in oneToNine without any where clause. The next step is to look at two digit numbers without repeating digits where n mod 2 == 0. So we write this:


> m2 = [n | a <- num
>         , b <- num \\ [a]
>         , n <- [a*10 + b]
>         , n `mod` 2 == 0
>         ]

This is a Haskell list comprehension and it can be read as "list ([) of n's (n) such that (|)...". Where the "..." is a series of conditions. The conditions are either pulling a value from a list (the ones with the left arrow <-), or are predicate statements. It will only use values that cause all the predicate statements to be true. We specifically have a being a digit. The second value b is using the list difference operator // which takes a list on the left and a list of values to remove on the right.

We will pause for a second and notice how similar this is to the mathematical statement:

If we run the code so far in ghci we can see that evaluating m2 gives us:


*Main> m2
[12,14,16,18,24,26,28,32,34,36,38,42,46,48,52,54,56,58,62,64,68,72,74,76,78,82,8
4,86,92,94,96,98]

Notice no digits repeat and all the numbers are even. Evaluating length m2 we see:


*Main> length m2
32

Since there are 9 digits (greater than 0) half of which are even (round down because 1 is odd) there should be four repeating digit even two digit number (22,44,66,88). Since we are not considering zero, there are nine possibilities for the first digit and four for the second (same as our repeats). This gives four times nine (36) minus our repeats (4) which equals 32. We now should be convinced that m2 is correct.

At this point we might be tempted to do the same thing for m3 but replacing our a to draw from m2. Instead we will capture the idea of that step in a more generic way. First we need a helper:


> toDigits a   = reverse $ h a
>    where h 0 = []
>          h a = a `mod` 10 : (h $ a `div` 10)

Back to ghci:


*Main> :t toDigits
toDigits :: (Integral a) => a -> [a]

This Haskell type shows that toDigits takes an Integral and gives a list of Integrals. It could be helpful to construct a sort of inverse to toDigits called fromDigits which could be defined as:


> fromDigits = foldl1 (\a b -> 10*a + b)

The details are not important, but it is helpful to notice that these two functions are nearly inverses of each other:


*Main> :t fromDigits . toDigits
fromDigits . toDigits :: Integer -> Integer
*Main> :t toDigits . fromDigits
toDigits . fromDigits :: [Integer] -> [Integer]

Here the . operator is composing the two functions.


*Main> fromDigits [1,2,3]
123
*Main> toDigits 123
[1,2,3]
*Main> (fromDigits . toDigits) 123
123

When we use fromDigits on a list with numbers larger than nine toDigits fails to be an inverse:


*Main> fromDigits [12,3]
123
*Main> toDigits 123
[1,2,3]
*Main> fromDigits [1,23]
33

This "failure" would still be useful in the case where only the first item in the list is greater than nine. In general it is nearly always instructive to try and construct an inverse function as it tells you quite a lot about your original function. You might observe similarities between fromDigits and n in m2. It turns out that there is another way to generalize n which we will come back to later.

Looking back at our definition of m2 we want to make that work for any step so we look for the parts that need to change. We know that we want to have a pull from the numbers from the previous step. And we know that we will be "modding" by a higher number each time. This implies two inputs to our step function which we will call ns and m respectively. We also have a problems that our helper function will fix for us. First the numbers from the previous step need to be split into digits for them to be subtracted from the pool of digits for b. So that becomes b <- num \\ (toDigits a). Now we can write our function down:


> step ns m = [n | a <- ns
>                , b <- num \\ (toDigits a)
>                , n <- [10*a + b]
>                , n `mod` m == 0
>                ]

We can test this out and see if it produces the next step:


*Main> let m3 = step m2 3
*Main> m3
[123,126,129,147,162,165,168,183,186,189,243,246,249,261,264,267,285,321,324,327
,342,345,348,369,381,384,387,423,426,429,462,465,468,483,486,489,528,543,546,549
,561,564,567,582,621,624,627,642,645,648,681,684,687,723,726,729,741,762,765,768
,783,786,789,825,843,846,849,861,864,867,921,924,927,942,945,948,963,981,984,987
]
*Main> length m3
80

I'll leave it as an exercise to the reader to determine if that is correct for m3.

We can now write out all the steps to get to m9 but we will now call them s1...s9 for steps one through nine:


> s1 = num
> s2 = step s1 2
> s3 = step s2 3
> s4 = step s3 4
> s5 = step s4 5
> s6 = step s5 6
> s7 = step s6 7
> s8 = step s7 8
> s9 = step s8 9

We can verify s9 is in fact the same result as the C# code gives:


*Main> s9
[381654729]

But we are not done yet! Writing out all the steps is repetitive. We will do a little bit of Haskell trickery to see how we can easily write this more simpily:


> s 1 = num
> s 2 = step (s 1) 2
> s 3 = step (s 2) 3
> s 4 = step (s 3) 4
> s 5 = step (s 4) 5
> s 6 = step (s 5) 6
> s 7 = step (s 6) 7
> s 8 = step (s 7) 8
> s 9 = step (s 8) 9

Here we used Haskell's pattern matching style of function definition to turn our s1 through s9 functions into a single function s that takes one parameter. All we had to do was add a few spaces and parentheses Now it should be clear how to collapse this down to two statements:


> s' 1 = num
> s' m = step (s (m-1)) m

We have s' 1 as the base case and s' m as the m^th case.


*Main> s' 9
[381654729]

Next time we will go further and look at some other ways to generalize this kind of problem.

Haskell

2010-03-31T07:17:00.002-06:00

I've been learning more about Haskell and came across this paper:

http://conal.net/papers/beautiful-differentiation/beautiful-differentiation-long.pdf

Reading it made some things start clicking...

Drawing programs

2010-02-28T13:07:00.003-07:00

I was playing around with some "Chrome Experiments" and came across a fun drawing program. I had to implement some aspects of a drawing program at work so I can appreciate something well made:

http://www.chromeexperiments.com/detail/sketchpad/

LaTeX Grapher

2010-02-23T09:26:00.009-07:00

In an effort to equip my wife with the finest tools for being a math professor I've started to tackle the seemingly (to me) gaping hole in math document creation of quality two dimensional graph generation. Quality means vector graphics. Document creation means no one-offs, but instead something that is maintainable for the lifetime of said document. These requirements lead directly to something language driven and with enough features (and ability to add features in a consistent way) to handle anything we throw at it. The initial result is this:

http://latexgrapher.codeplex.com/

It's not ready for binary releases yet, but it will get there. Right now it requires .NET 3.5 (probably SP1). It can be compiled by Visual Studio C# 2008 Express Edition (free download). I will probably make a Silverlight version eventually, then you can use your web browser (and Silverlight plugin) to make graphs from Linux, Mac, or Windows. There will also eventually be a command line version that can be integrated into your desired build process.

What can it do now you ask? Here are a few examples.

Output	Code
	`View (-3,-3),(5,5) f(x) = x^3/20 + 2 Shade f,(-2,3)`
	`View (-1,-3),(6,4) f(x) = x^2/5 \| x < 2 3 \| x == 2 x^2/5 \| x < 4 -(x-2)^2/2 + 3 \| : x = 2,4`
	`View (-1,-1),(11,11) f(x) = NaN \| x < 0 3 \| x < 1 3+2*floor(x) \| x < 4 10 \| : x = 0,1,2,3,4`
	View (-10,-10),(10, 10) > r = 5 > n = 5 > x0 = 2 > y0 = 3 > a(t) = tpi/(2n) > fx(t) = rcos(a(t)) + x0 > fy(t) = rsin(a(t)) + y0 > gx(t) = -(fx(t) - x0) + x0 > gy(t) = -(fy(t) - y0) + y0 Line (gx(0),gy(0)),(fx(0),fy(0)) Segment (gx(1),gy(1)),(fx(1),fy(1)) Line (gx(2),gy(2)),(fx(2),fy(2)) Segment (gx(3),gy(3)),(fx(3),fy(3)) Line (gx(4),gy(4)),(fx(4),fy(4)) Segment (gx(5),gy(5)),(fx(5),fy(5)) Line (gx(6),gy(6)),(fx(6),fy(6)) Segment (gx(7),gy(7)),(fx(7),fy(7)) Line (gx(8),gy(8)),(fx(8),fy(8)) Segment (gx(9),gy(9)),(fx(9),fy(9)) g(x) = x

LaTeX in blogger.

2008-09-30T12:16:00.003-06:00

I found a nice website that allows you to put LaTeX inline with blog posts and other things. For example:


 X = \{ x | x \in (1, 2) \}

All I had to do was add some javascript to my template.

Glacier Animal Identification Updates

2008-07-20T20:27:00.008-06:00

It's that time again! Rebekah's family was in town a couple weeks ago and we took them to Glacier and really hit the jackpot for animal sightings. I thought I would start out with the new identifications by updating some of the animals already featured in this series:

Bighorn:

Rebekah was able to catch a video of some young bighorn practice their head crashing skills. We also got the chance to pick up a bighorn skull in the discovery center (there were lots of fun things in there). Their horns are really heavy, I'm glad they don't like to headbutt people.

Marmot:

As always the Marmots were not shy for attention. Once we reached Cracker lake we had one very curious hoary Marmot walk right in front of me posing every few seconds.

But not all the attention went to the marmot:

Moose:

We saw several moose with antlers this time on our scouting trip before Rebekah's family came.

Then when they came we saw one as we were hiking on the trail.

I'll have more updates soon with new animals!

Out of the blue.

2008-05-22T13:47:00.003-06:00

Here is another stack from When Do I Get a Brownie:

“Drop the fire sword”

Time is of the essence when it comes to brownies.

2008-05-15T19:19:00.003-06:00

When do I get a brownie?

2008-05-15T11:27:00.005-06:00

So awhile back we were inspired by a blog and since the game either didn't have a name or we didn't remember it, we decided to call it “When do I get a brownie?” (A quote from Rebekah's mom). The game is kind of a visual telephone game where a phrase is interpreted as a picture, which is then interpreted again as a phrase. To play, each person (6-9 people is best) starts with as many blank squares of scrap paper as there are people. Each person then writes a phrase on the top piece of paper and passes their whole stack to the next person. When a player gets a stack they read the phrase and place that piece of paper on the bottom of the stack and draw a picture that represents the phrase. The stack is passed again and the player that gets the picture looks at it and then places it on the bottom of the stack and writes what phrase they think the picture represents. The game then ends when the stack gets back to the original phrase writer. Once everyone is finished the stacks are shared and much hilarity ensues.

While we were making Triple Chocolate Coffee Bean Cookies the other day we played “When do I get a brownie?” with four players passing the stacks two rounds (we also added a fifth stack to make it a little harder to remember which stack was the one you started with). Here is one of the stacks:

Cindy started out with “Wet socks are bad luck”.

Clara followed with some lovely wet socks (most certianly unlucky on their own) and a crossed out four-leaf clover.

Rebekah interpreted this as “Gym socks dripping with sweat are not a good luck charm.” A reasonable interpretation.

I then drew a charm necklace featuring a four-leaf clover, rainbow, gym socks (crossed out and dripping), fuzzy dice, and a pony (quite the charming necklace). My attention to detail here caused quite a backlog.

Cindy then interpreted that as “Charm bracelets should never have socks” (a truism I'm sure).

Clara then took the quick route with the impression of individual charms (one slightly detailed towards a four-leaf clover) an implies arrow and a no socks sign. I think I would have interpreted this one as “Beads imply no-sock-wearing hippy”.

Rebekah instead thought that that meant “Charm bracelets are not gym socks”.

I was still in the mindset of “necklace” so I quickly drew a charm necklace that only a geometry teacher could love not equaling socks. I'll post some of the funnier stacks so that all can enjoy.

Glacier Animal Identification VI

2008-03-16T14:27:00.016-06:00

First let me say that these animals are most certainly wild and should never be directly approached. Always respect the animal's space.

While their name is misleading as they are part of the antelope rather than the goat family, the Mountain Goat is the certainly deserving of the mountain part of their name. They pretty much ignore other lifeforms and go about their business knowing that they could climb straight up a near by cliff and get out of harms way if needed. All the rest of the animals seem to know this too so they don't bother. Some eagles are a threat to the babies and the nannies will defend against such attacks. Here are a group of goats by the trail waiting out the wind:

When they are not getting blown around in the wind they seem to be fond of posing for pictures. This one even cooperated in trying to make the ledge look as precarious as it could:

Baby mountain goats are quite possibly the cutest creatures in Glacier (Rebekah has told me this).

Much of our time in Glacier is spent documenting baby mountain goats and their cuteness.

We have much much more, but really you just have to see them in person.

Update:The videos do not show up even though they show when previewing them in the editor. I assume this is a temporary issue that Blogger will figure out soon.

Glacier Animal Identification V

2008-03-16T14:09:00.004-06:00

Bighorn Sheep are pretty much sheep with big horns. Their bodies are pretty big too, but they tend to be pretty docile creatures. They can get away for most any predator by using their excellent climbing skills and the rocky terrain.

We have some other pictures of bighorn but they are not on this computer, so we will post them later.

Where did July go?

2007-08-08T21:03:00.001-06:00

We had an eventful July filled with a bunch of stuff. The Agony ride and the weeks leading up to it were excellent and fellowship filled. Our friend Jamie and my brother Travis were both there:

Rebekah had a crash in the morning (she hit my wheel from behind and went down), but she was fine and the scrapes and bruises are nearly gone already. After getting patched up on the road she insisted on riding the rest of that leg and went on to ride 171 total miles. Our new bikes performed excellently and Rebekah and I have been on three rides since.

Our new software at work is finally starting to take shape as it continues to occupy most of my time.

While we were in California for the Agony ride our plants all grew very nicely. We have two new avocado seeds sprouting and our existing plant has foot long leaves on the top. This weekend we are going back up to Glacier National Park to ride the Going to the Sun Road again.

Going-to-the-Sun

2007-06-26T10:28:00.000-06:00

This weekend we had an amazing trip to Glacier. We decided to combine our love of Glacier with our Agony Ride training and the result was taking arguably the most spectacular bike ride in the United States.

The starting point was at 3500 feet at the Avalanche campgrounds. We left with a friend from our church who didn't know what she was in for but stuck it out and had an amazing ride on her 20 year old Mountain Bike that weighed more then Rebekah and my bikes combined. We ended 14 or so miles later and at about 6000 feet (the whole Going-to-the-Sun road is still not open). The next picture is the view we had from the top of our ride (taken the previous day).

It was quite the sight to look back and know that we had started at the bottom of the valley in the right of the picture.

Riding back down was fast and only a little over the speed limit (okay 24%). So we didn't poke our heads up too much for views like this:

Agony Ride

2007-06-18T21:29:00.000-06:00

As many of you already know I've been blessed to be a part of Christian Encounter Ministries in some way for my whole life. Every year they have a major fund raising event called the Agony Ride where willing victims get sponsors to support them in their effort to ride a bicycle as far as they can in 24-hours. Here is what they have to say about the event:

Christian Encounter Ministries is a domestic mission organization with the unique purpose of discipling and counseling troubled young people. These young people come to CEM with major damage from families, society, and/or their own poor choices. As a domestic mission, we are dependent on the loving generosity of mission-minded ministry partners. Our annual Agony Ride is an opportunity for our partners to participate in this vital mission. Some of our staff and approximately ninety other riders attempt to push their limits by riding as far as they can in 24 hours. The Agony has become more than just a personal challenge; it now stands as a pointed demonstration of Christ's love for our students as well as a witness for Christ in the communities of the Sierra Valley.

Rebekah and I are going to be riding this year (the 25th anniversary of the ride). While it is easy to see the obvious benefit of the ride in raising monetary support for the ministry what isn't obvious is the other ways that the ride supports the ministry. The students that are enrolled in the program are participants in the massive effort that supports the riders and keeps them safe and well fueled. While they are watching they get to see a working picture of the body of Christ, with each member doing their part to supporting those around them.

I'll post some more in the coming weeks about our training for the ride.

Glacier Animal Identification IV

2007-06-18T20:52:00.000-06:00

The largest squirrel you will see in Glacier is not the illegally fed Columbian Ground Squirrel. It is instead the largest of North American ground squirrels, the Hoary Marmot.

Named for their speckled gray color, the gargantuan rodents enjoy life on the talus slopes.

As you can see by how close we took our photos, they are pretty much oblivious to humans. We did not intentionally approach that closely to them, they just popped out of the bushes right in front of us and went about their business. The one in the first two pictures might have been leading us away from its little baby marmot which, because of the mother's great distraction skills, we didn't notice in time to get in the picture.

They have a distinctive whistle that can often be heard when hiking near the tree line.

While they may not qualify as Rodents of Unusual Size, they are certainly Squirrels of Unusual Size.

More Deer

2007-06-18T20:41:00.000-06:00

We went to Glacier again this last weekend and were pleasantly surprised to have Going-to-the-Sun road opened up most of the way for the first time this season while we were there. It was beautiful as usual, but we were unable to get pictures of any newly identified critters. There were a lot of our deer friends sporting their fresh new summer coats.

Glacier Animal Identification III

2007-04-10T21:36:00.000-06:00

Seeing as it is Easter time, I thought the next animal to be highlighted in our series would be the Pika. In the same order as rabbits, these little mountain fur balls industriously chop and dry hay during the warm months, then stow it in the talus slopes they call home. In the winter they nibble on their hay and I suppose catch up on their reading. While hiking in Glacier, you will hear their distinctive chirp long before seeing one. A sound clip of their call can be found here. We caught up with the fellow pictured here while hiking the Highline Trail from Logan Pass headed to the Granite Park Chalet. It only stood still for one photo, then scurried off. We did see some of its drying hay bales, but I didn't take any pictures of those :(.

The full picture is available here.

Glacier Animal Identification II

2007-04-01T17:42:00.000-06:00

Continuing with the animals we have identified in Glacier, we have deer. The first time we went to Glacier we were totally amazed by the exciting creatures unique to the alpine habitat. We had just spent the day with mountain goats wandering around mere feet from us and were about to head down from Logan Pass when we were asked by a couple to take their picture. They seemed excited to be in the park and they lady told us that they had just "seen a deer from their car". A whole live deer? from your car!?! They have those here? No, no, I just smiled my head and nodded wondering what her reaction would be to a smiling mountain goat walking the trail with her (hopefully they did eventually get out of their car). All that to say, yes, they are just deer, but deer are still really cool to see.

First we have the mule deer (at least we are pretty sure that's what it is). Apparently they like to patrol the trails in the summer.

Back up the same trail we encountered an antler-adorned whitetail deer.

The whitetail has a much different color and thicker coat in the winter.

They are clearly named by the very whitetail which they flare as an alarm when startled.

In the last picture the lookout deer (we are guessing the mother) decided we were a danger (even though there was a fast flowing creek between us). The child deer was seemingly not responding quickly enough, causing this response from the mother:

Glacier Animal Checklist

2007-03-29T23:55:00.000-06:00

Rebekah and I just got back from visiting Glacier for the first time in the spring. While there wasn't enough snow low enough for us to go Cross Country skiing, we still saw some spectacular mountains and fun wildlife. This inspired me to start a series here on the blog with all the animals we have seen and identified in Glacier. At first I was going to do it in one post, but there is too much to cover, so you will get several.

To start it off, we have the ubiquitous greeter of all park goers and universally loved food source for all carnivores bigger than it, the Colombian Ground Squirrel.

They are always alert and looking for attention or the nearest predator, whichever comes first.

Oh and a bonus picture so you can see the tail.

2007-03-26T17:32:00.000-06:00

Rebekah conquered Big Mountain (she even did a black diamond).

2007-03-23T15:06:00.000-06:00

Here is an example of a healthy cat scan.

2007-01-10T12:07:00.000-07:00

We got a new camera this Christmas. I couldn't decide between the Canon G7 and the S3 IS. I never found a store that had a G7 that I could play with and there were several things that I thought I would miss about my G3. One of the biggest ones is the flip out screen. I don't really care too much that the screen on the G7 is bigger, it is much more about being able to position it, and tuck it away where it won't get scratched. The draw backs to the S3 included that both viewfinders are LCD's, so there is no "analog" aiming device. The biggest problem with this is that even if you get the shutter lag really good, you have the added video lag. Regardless either one was going to be able to take pictures much more quickly then the G3 does. Neither one had support for a remote control which will be much missed. In the end we got the S3 and I have been quite happy with it so far. Other detracting issues include no built-in recharging (no power adapter at all), no info display on the top like the G3 had.

Now the plusses. The zoom is huge, and the image stabilization (optical) works (you can't expect miracles of course). The picture above was taken from the back seat of my in-law's car (granted it is a Metro so it is the equivalent of the front seat in an SUV) at full zoom while zooming down the Pennsylvania turnpike at 60 mph. Pennsylvania is not know for smooth roads either, but I don't remember any big bumps =). Maybe my hands were just rock steady at that early in the morning (sunrise in Pennsylvania when we had come from California the day before).

The other thing I like is the improved video quality. I will have to take advantage of that...

2007-01-10T11:35:00.000-07:00

This is from Rebekah driving the Yaris for the first time. The rest of the shots I took were kind of blury. We are still getting used to having the instruments in the middle, especially at night when we think our lights are off.

2007-01-05T12:44:00.000-07:00

We got our new car last night. More to come...

2006-12-27T22:06:00.000-07:00

We have a Christmas blog:

http://yateschristmas.blogspot.com

Have fun.

2006-12-14T16:13:00.000-07:00

I ran into an annoying bug with GDI+ yesterday.

http://connect.microsoft.com/VisualStudio/feedback/ViewFeedback.aspx?FeedbackID=245730

Somebody didn't check their edge conditions :(.

2006-12-11T21:15:00.000-07:00

At the previously mentioned Christmas party I spent most of the time chatting with a pharmacy grad student about math education. He felt ripped off by the educational system because it did not make math interesting enough and now he doesn't readily have the mathematical tools at hand to most effectively do his work. So we mostly ended up talking about what we wished we had different about our math education. One of the things that struck me was that nearly all of the exciting things that happened in my pre-college math education were from my own investigation/curiosity and if it involved a teacher it was in a group of three or less.

I've talked to Rebekah before about how neither of us can remember where we learned many core math concepts that we do not struggle understanding, but we can clearly see the way that math we learned later built upon that foundation. From my perspective now if I see someone struggling to understand addition or multiplication I revert in my head to an axiomatic basis for the natural numbers that asserts the number one and successor that is also a number. Should schools teach math and numbers at that level until the kids get it then move on to higher concepts like addition and multiplication? After that the concept of inverses so that division and subtraction come for free? Really shouldn't students understand our place value number writing system inside and out as soon as they dare to count passed nine? Can this be avoided with such tactics?

One thing that I think helped Rebekah and I was an ability to have negligible regard for math application in the course of learning. Neither of us needed justification to study math beyond obtaining a deeper knowledge of math. Word problems only become annoying in that they bother to contrive a situation that must be abstracted out to get to the real problem. Maybe we were motivated at times in the thrill of solving a puzzle, but what makes a puzzle interesting is the beauty or wit of the solution.

So how was I inspired to recognize and appreciate the beauty and wit of mathematics? Sadly I don't remember.

2006-12-10T16:17:00.000-07:00

Rachel, my sister-in-law, doesn't think I post anymore so she is going to unlink me! I have an excuse though... things at work have been really busy these days. We have had a lot to do with our new version of the software (which is really cool and I can't wait to have it all working) and the owner of the company has been out as his wife is having some critical medical problems. Sandy's swift recovery is in our prayers.

For Christmas we will be traveling down to California and then after Christmas we head east to Pennsylvania. When we get back to Montana we will have our new car (Toyota Yaris liftback). Currently it is somewhere on a boat in the Pacific ocean.

Last night was the Math department Christmas party. We made sugar cookies shaped like trees, candy canes, gingerbread women (they have skirts, so maybe they are Scottish gingerbread men), numbers, a minus sign, a less then or greater then or hat symbol, some dots, and a T-Rex. Rebekah puts lemon in the mix for flavor and thinks that that is unique (I don't know if it is but it tastes good).

Our house and cat are being house sat by one of the youths from our church. She visited today with her two kittens which inspired some territorial responses by Mobius, but they seemed to get along well enough. As they were leaving one of the cats didn't like our neighbor's dog and sprinted (cheetah style) back through the hose and across my left second toe. I'm ok though, just a little bleeding, nothing like this.

This afternoon we made fudge (chocolate and peanut butter). Rebekah likes to make "finals week fudge" for her students to consume. I'll make her post about it on the cooking blog.

In sports news, I've been too busy to do anything active recently (we should be getting out on the snow sometime soon), but both Wheaton's soccer teams were in the NCAA III finals with the women delivering a resounding victory and the men loosing, but making for an outstanding finish to Joe Bean's career.

2006-10-26T01:20:00.000-06:00

So I decided to move the whole blog thing to blogspot. Next week I will be at the IFAI expo in Atlanta. Hopefully I will get to see some friends from the time Rebekah and I spent there. Recently I got a shiny new laptop (no really it's shiny), an so the old one is going back to the corporate headquarters so I'm reinstalling everything on the old laptop. I started by trying to install Windows Vista RC1 on it, but apparently Gateway hasn't updated their BIOS for Vista yet. So I loaded XP and stuck in the Gateway drivers CD and it took over and started installing and rebooting after each install. It said in large print, "don't push any keys or buttons", so I didn't but it still required me to click on my user name ever time after it booted. Now I feel like John Locke, existing to push the button...

Now it's bedtime as XP SP2 is installing.

2006-03-29T09:43:00.000-07:00

Mobius' latest film is an action thriller where she goes Stallone on all her favorite baddies. "Hey plastic bag, you're the disease and I'm the cure!" You can tell when she is about to bat crime off the map by the size of her pupils.

2005-11-25T14:08:00.001-07:00

Yet another shot of the swing.

2005-11-25T14:08:00.000-07:00

Rebekah vs the Spiky Frost.

2005-11-25T14:07:00.001-07:00

Spiky branches

2005-11-25T14:07:00.000-07:00

For the first time in a week the fog has lifted and we can see the mountains, No sun or sky yet, but at least we can see more then 100 yards. There was some wierd spiky frost on everything this morning too. It seems like each day this last week we have had a differend variation on frost each mornings. Sometimes, little white dots, sometimes snow frozen in clumps, and now the spiky stuff.

2005-11-06T22:52:00.000-07:00

Rebekah got me into Foxtrot (www.foxtrot.com) awhile ago and I have continued to look it up everyday for at least a year now. Having a Tablet PC from work now I couldn't resist doing this Sunday's puzzle offering from Jason.

2005-11-04T21:55:00.000-07:00

we were designing the new interface to the X 5
and we ended up not really using it for anything.
But that is about to end! the tablet has been
Sent to me in the north office to do with what
I will." />

2005-10-16T17:36:00.000-06:00

This last week we have had John Michael Mugerwa in Missoula sharing about his ministry to AIDS orphans in his home country of Uganda. He oversees a boarding school orphanage called "Destiny School" about 25 miles outside of Kampala. It has been a true blessing to have him with us. This morning he spoke of how important prayer is to the church in Uganda. Prayer is central when they gather together and is seen as the battleground for the work of the church. John Michael pointed out that Jesus won the battle in the garden of Gethsemane on his knees where he uttered the words "not my will, but yours be done." (Luke 22:42) which directly led to his salvific work on the cross. He also said that the armor of God is put on to do battle in prayer. What a picture, Righteousness, Truth, the Word, faith, and Salvation surrounding us as we pray for God's work on earth. Is there any other way to go into battle?

It fills me with joy to hear John Michael's stories. Two hundred of the children come from Muslim families and several of those family members are now Christians at John Michael's church because they have visited the children and seen the love that is poured out on these kids. Some of these children were left on the street because nobody cared for them after there parents succumbed to AIDS. As John Michael put it, Islam cannot stand up to the practical love of Christ. When children come to Destiny School they are no longer orphans because they have been adopted into Christ's family.

2005-09-25T13:46:00.000-06:00

Snow on Lolo. Usually the view is the opposite, clouds on Lolo and none elsewhere, but this morning the clouds framed in the peak while the sun shined brightly on the fresh snow.

2005-09-05T23:03:00.000-06:00

A smoky day on Saint Mary Peak. Although we couldn't see the bottom of the Bitterroot valley from the top of the mountain it didn't stop us from having a lovely hike.

2005-08-05T08:26:00.000-06:00

Rebekah and Mobius studying

2005-08-04T22:57:00.000-06:00

New blog and the Agony Ride.

Rebekah and I have started a cooking journal found at http://storycooked.blogspot.com/. It should be interesting.

We recently got back from the Agony Ride in the Sierra Valley north of Lake Tahoe. The Agony Ride is the fund raiser that Christian Encounter Ministries does every year and is called such because you ride a bike for 24 hours, or at least close to that amount. I had a good ride, 169 miles. It started well and it ended great (let's not talk about the middle). Rebekah did excellently too with 150 miles (and she didn't think she would make 100). All around it was a good time to see friends and help raise support for a wonderful ministry.

In other news, it's smoky out, so we don't have the house opened up and it is hot.

2005-06-29T21:02:00.000-06:00

It is a month until the Agony Ride!

Rebekah and I are going to ride in the 24-hour bike ride fundraiser for Christian Encounter Ministries this year known by those who ride it as the Agony Ride (it's also named that). The ride goes back and forth between three different SAG (Support and Gear) stops with Loyalton CA in the middle. Riders raise support per mile, I hope to ride 200 miles this year.

Christian Encounter Ministries is a residential counseling ministry for youth from ages 16 to 25. I did an internship there after college (if you look closely on their website you will see the wonderful interns I worked with). The ministry there is two fold, helping the kids and teaching the interns about ministry.

If you want to know anything about CEM, or the ride, send some email!

2005-05-30T14:08:00.000-06:00

We drove by some Not-so-big-horn sheep on our way back from visiting family in Boise. Well hopefully they will grow up to be really-big-horn sheep, that is if they learn to stay away from the highway. They were so close I could smell them. Up on the hill was a mama sheep with her little lamb. At least she had some sense.

Next weekend we are going up to Glacier National Park, expect more pictures.

Later

2005-04-18T19:41:00.000-06:00

We ventured to the National Bison Range last Saturday. It was beautiful, but there were no baby Bison as we had hoped. Getting out of the house into the open range is always welcome (even if you have to stay in the car so as to not become a Bison welcome mat).

The mountains in the background are the Mission mountains which form the east side of Flathead lake (at least they do a little more to the north).

2005-03-17T23:15:00.000-07:00

It's March Maddness time.

2005-03-04T10:59:00.000-07:00

I put together some photos I took while I was building a dresser last weekend:

http://home.bresnan.net/~fryguybob/blog/desser_0002.wmv

If anyone wants a Mac friendly version just let me know I think I can swing it.

Ryan

p.s. Fixed url

2005-03-03T17:23:00.000-07:00

Our Montana neighborhood.

2005-02-27T21:31:00.000-07:00

Rwanda is forgotten Again.

Is a truly excellent film about self-sacrificial love in the midst of unthinkable horror not worth recognizing? Oh wait, I forgot it's the Oscars®. We need to celebrate American angst, the American dream. How a single person can fight against the odds and come out on top (Howard Hughes? Ray Charles?). I guess the thousands of lives that Paul Rusesabagina saved don't make the cut. He didn't touch millions with his airplanes or his music (no offence to the late Howard Hughes or the late Ray Charles). Mr Eastwood has had such a long career though, he deserves to be recognized. Black Hawk down at least had good enough sound, dispute it being about Africa American troops. Riddly Scott gets a nomination nod, but again he had such a distinguished career. Besides Bruckheimer was behind it so it's safe to nominate.

Was Hotel Rwanda too artsy? Had it won too many awards already and didn't need any more? Did they think, "well we have room for only one hero award and by golly that Incredibles was incredible."

The AP writes:

The 77th Oscars were another heartbreak for Scorsese, the man behind "The Aviator," who lost the directing race for the fifth time.

Boo hoo, is that the bigest snub story they could find?

200,000 killed in Rwanda while America turns its back and now the best American film and the best foreign film by American estimation both carry the theme of assisted suicide? Fitting.

2005-02-24T14:51:00.000-07:00

As a fan of NPR's online Music show All Songs Considered, I knew that they were featuring a live webcast of Wilco tonight. Last time it was Bright Eyes and there were a couple of other bands that played first, so naturally I was wondering who would be playing with Wilco. Somebody hip? edgy? Maybe another Chicago band waiting to be discovered by the rest of the world? Perhaps the one band that delivers the most efficient rock and roll available? Yes you guessed it, the Detholz are opening for Wilco, live webcast at 8:30pm eastern:

http://www.npr.org/templates/story/story.php?storyId=4502847

From the NPR website:

Detholz (pronounced "death holes") play a mix of power pop, punk and new wave. The band's members met in 1996 at what they call a fundamentalist Bible school and now perform "outsider" songs with "Christian kitsch."

Lets hope they play Monolith of Death,

Yates

2005-01-20T14:58:00.000-07:00

Last night at bible study we were looking at Ephesians 5 which begins:

Therefore be imitators of God, as beloved children. And walk in love, as Christ loved us and gave himself up for us, a fragrant offering and sacrifice to God. (ESV)

We are to imitate God and we have an example, Christ (who is God with us). Profound.

I hope to in this blog explore what it means to be an imitator of God with Christ as my example in a technologically filled world.

peace.