Woke up to a wet day, and no possibility of a hike, so Barbara and I decided to drive from one end of town to the other just taking in the storefronts and the ambiance. Grand Lake looks good with mist rising off the streets and all glittery wet.
Before checking out we stood - out of the rain - and watched the lake. Even with the rain coming down there were still paddle boarders. And they appeared to be having fun.
Decided to go over Berthoud Pass and maybe eat lunch somewhere along the way. As always this friendly little pass was delightful. The climb was through gorgeous weather and we could see forever. The views along the way were of cirques and glacier carved mountainsides, pine and aspen forests, distant lakes and cloud-covered peaks. Made my heart sing (if you thought of Wild Thing just now so did I).
Went into the town of Empire to visit Jenny's Diner. Our grandson-in-law's parents own the wonderful little establishment and we have never met them. They were gracious and fun and offered us a free lunch but we wanted to get through Denver before rush hour. The diner is situated in an old antique shop (Did you know that potato chips used to come in a can?). Took some pictures of the town and the diner and moved on.
Got home around noon feeling refreshed and promising ourselves we would hit Rocky again as soon as possible.
So this is the end. If you have never visited to the west side (some call it the wet side) of Rocky Mountain National Park, don't put it off. Elves dance there.
So long.
Sunday, August 24, 2014
Saturday, August 23, 2014
Rocky Mountain National Park Day 4
Woke up this morning, looked out the widow and saw a mist over the entire lake. Looked five minutes later and Grand Lake and all the mountains behind it were gone. We were inside a big cloud. Very cool!!!
When it cleared Barbara walked again down to the rapids. I wrote for about half an hour. Very peaceful.
After breakfast we went to Bowen-Baker trail. I do believe this is my new favorite. A gradual climb up into a national forest. Very few people. Very green and lush. The sounds were indescribable - water in the river, birds in the canopy, even the growl of something I think was a bear. The path was soft underfoot and the company (my wife Barbara) was exceptional.On the way back we saw a moose along the road (with calf).
Went back and had lunch overlooking Grand Lake. Very nice.
In the afternoon rented a kayak and went out onto the lake for an hour.
Barbara took a nap. I took my computer out onto the tables overlooking the lake and wrote for about an hour. Soul-restoring time.
Went back into the park in late afternoon, herds of elk with bunches of babies, more moose, and a particularly beautiful deer with velvet antlers. The deer did that thing that gazelles do, just bounced across the road and off into a meadow. Never walked a step.
Watched the lake as the sun went down and the stars came out.
Good night.
When it cleared Barbara walked again down to the rapids. I wrote for about half an hour. Very peaceful.
After breakfast we went to Bowen-Baker trail. I do believe this is my new favorite. A gradual climb up into a national forest. Very few people. Very green and lush. The sounds were indescribable - water in the river, birds in the canopy, even the growl of something I think was a bear. The path was soft underfoot and the company (my wife Barbara) was exceptional.On the way back we saw a moose along the road (with calf).
Went back and had lunch overlooking Grand Lake. Very nice.
In the afternoon rented a kayak and went out onto the lake for an hour.
Barbara took a nap. I took my computer out onto the tables overlooking the lake and wrote for about an hour. Soul-restoring time.
Went back into the park in late afternoon, herds of elk with bunches of babies, more moose, and a particularly beautiful deer with velvet antlers. The deer did that thing that gazelles do, just bounced across the road and off into a meadow. Never walked a step.
Watched the lake as the sun went down and the stars came out.
Good night.
Friday, August 22, 2014
Rocky Mountain National Park Day 3
I didn't think anything could top the wonderful day Barbara and I had yesterday but today was almost magical. From Adam's Falls up in the mountains (Grand Lake sets kind of in this bowl/valley) at the north end of the lake, a cascade sends rapids down into the town. In fact there is this really cool restaurant cleverly called The Rapids. Barbara is enamored with these waters and walked up to where they roar down the mountain (while I slept in-clear to 7:30).
After that and breakfast overlooking a misty lake, we hiked the Shadow Mountain Lake trail. Shadow Mountain Lake is the next lake over from Grand Lake and you can actually canoe from one to the other. The trail skirts this lake for the entire length of it, through pine and aspen forests and up and down rocky hillsides. It had rained the night before and we expected mud but the trail was in good shape and we kicked its butt.
After lunch we headed into the park proper and went to Green Mountain Trail. A steep climb toward an area called Big Meadow (I have it on the best of authority this is a relatively large meadow). On the way back, we were coming down the path when not twenty feet away was a mama moose and her baby. Cool as this is to see, it is also crazy dangerous. Moose are nuts as it is. One long time resident has lost two friends to moose in the last 2 years - and these weren't the crazy tourist types (like me), these were just unlucky folks who stumbled upon a moose who took it into its head to trample them to death. The most dangerous of all is a mama with her baby. Last year we were parked up at a place called Beaver Ponds (which is a flat out misnomer since no beaver live there). While we were sitting in our car that evening, a mama moose came into the parking lot. For some reason we had a picked up a flyer at the ranger station called, 'Warning Signs a Moose is About to Attack'. While we ticked off the signs:
- Ears turned forward, check
-Nostrils flared, check
-Walking slowly toward you, check
well you get the picture, the mama made it plain she was going to run at our car. Shoot we were in a car and figured the moose would get hurt more than we would so I honked. The mama startled and walked away. This year we had no car, just a ridiculous proximity to this pair of animals. We had no choice but to keep going and the mama ignored us and kept chomping on the wet grass. By the way both she and her offspring were magnificent.
After we came down off Green Mountain Trail, we saw a pair of deer, including a fawn all spotted and tiny. In fact, after seeing the moose, even the grown deer looked petite. We also saw a herd elk right along the roadside. They ran back and forth across the road like they had left their keys behind or something. They were so fun to see.
Spent the evening walking through town. Went over to the bandstand in the center of town where a fella was playing. I guess acts play there every Wednesday night. This guy even played one of my favorite songs, 'The Weight' by The Band. If you don't recognize the title it is the song that begins with "Pulled into Nazareth, was feeling about half past dead..."
Good night all.
Wednesday, August 20, 2014
Rocky Mountain National Park Day 2
The previous day we passed over Trail Ridge Road, a continental divide highway that reaches a height of 12, 000 feet. We dropped down into the magical town of Grand Lake (The Soul of the Rockies), which sits on the western edge of Rocky Mountain National Park. On the way over, Barbara saw a huge herd of elk. An auspicious beginning.
Now here's the deal about Grand Lake and the western side of Rocky Mountain Nat'l Park. The place is moose central. Whenever I go I always see a buttload of moose: Daddy's with beards and large racks, females with long spindly legs and long faces, and babies (calves) who are cuter than kittens. Another deal about Grand Lake. It has this really grand lake, actually two if you count Shadow Mountain Lake. And why wouldn't you? From the deck of our motel we could see both lakes and for my money, Grand Lake is every bit as beautiful as Lake Como in Italy
Now I am 63 and when I turned 62, I paid $10 and purchased a lifetime pass to every National park in the US. That's right free for the rest of our lives!! The intention was to make our way into the park, see a bunch of wildlife and go ooh and aah a lot. So after eating breakfast overlooking our beautiful lake, we made for one of our favorite spots, Adam's Falls. A short hike out the end of Grand Lake, it really isn't in the park but it sure looks like it is--plus we saw a family of deer (buck, doe, and spotted fawn). Then we needed more of a hike so we headed to a spot which we call Moose Creek (not because that is it's name but because two years ago we had a dangerous liaison with a momma and baby moose). The creek itself is worth the drive but we always take an extended hike along the water and this time we ended up leaving the park and heading into national forest land-beautiful beyond compare. On the way back we saw a fox, another pair of moose-baby and mom-and a herd of elk( about 30).
We finished the night with a meal at the wonderful Grand Lake Lodge. From there we could dine and watch the sun set over Grand Lake.
Good night.
Now here's the deal about Grand Lake and the western side of Rocky Mountain Nat'l Park. The place is moose central. Whenever I go I always see a buttload of moose: Daddy's with beards and large racks, females with long spindly legs and long faces, and babies (calves) who are cuter than kittens. Another deal about Grand Lake. It has this really grand lake, actually two if you count Shadow Mountain Lake. And why wouldn't you? From the deck of our motel we could see both lakes and for my money, Grand Lake is every bit as beautiful as Lake Como in Italy
Now I am 63 and when I turned 62, I paid $10 and purchased a lifetime pass to every National park in the US. That's right free for the rest of our lives!! The intention was to make our way into the park, see a bunch of wildlife and go ooh and aah a lot. So after eating breakfast overlooking our beautiful lake, we made for one of our favorite spots, Adam's Falls. A short hike out the end of Grand Lake, it really isn't in the park but it sure looks like it is--plus we saw a family of deer (buck, doe, and spotted fawn). Then we needed more of a hike so we headed to a spot which we call Moose Creek (not because that is it's name but because two years ago we had a dangerous liaison with a momma and baby moose). The creek itself is worth the drive but we always take an extended hike along the water and this time we ended up leaving the park and heading into national forest land-beautiful beyond compare. On the way back we saw a fox, another pair of moose-baby and mom-and a herd of elk( about 30).
We finished the night with a meal at the wonderful Grand Lake Lodge. From there we could dine and watch the sun set over Grand Lake.
Good night.
Rocky Mountain National Park Day 1
Okay, reader here's the deal. My wife, Barbara and I traveled a few hours from our home in Colorado Springs, CO to venture into--what I believe--is the premier National Park in the state of Colorado.
Day 1 Estes Park and the Bear Lake hike to Emerald Lake. We came into Estes Park, a sweetheart of a town on the Eastern side of Rocky. This is the more traveled side of the park and in the heart of the summer and in the fall elk rut gets somewhat crowded. We are here in August and its delightful. We go into Rocky and immediately take what is called Bear Creek corridor up to one of the more popular sites, Bear Lake. This small lake is the jumping off point for most of the trails on the eastern portion of Rocky. Today, we choose the triple lake hike of Nymph, Dream, and Emerald Lakes.
Nymph Lake - the smallest of the triple lakes it is covered by lily pads in summer and in July these bear a beautiful the yellow flower.
Dream Lake - the largest of the lakes on this hike, it is a favorite spot for fishermen. A few years back I came to Rocky in the company of two fly fishermen (I wrote while they fished, 12 hours whew!). Today when Barbara and I arrive there is a conclave of Hebrew students (forelocks, yarmulkes, and black derby hats). Their unexpected presence in this western setting was underlined by the fact that most of them had wonderful New York accents.
Emerald Lake - every time I visit this last gem on the triple string I'm always tempted to think 'this is a place where elves would come to dance.' On the trail to this final lake we pass a waterfall and a wetland meadow alive with butterflies. At the lake we meet folks from all over the country, including a married couple from North Liberty Iowa. We come to find out they often frequented a restaurant owned by my wife's brother.
Day 1 Estes Park and the Bear Lake hike to Emerald Lake. We came into Estes Park, a sweetheart of a town on the Eastern side of Rocky. This is the more traveled side of the park and in the heart of the summer and in the fall elk rut gets somewhat crowded. We are here in August and its delightful. We go into Rocky and immediately take what is called Bear Creek corridor up to one of the more popular sites, Bear Lake. This small lake is the jumping off point for most of the trails on the eastern portion of Rocky. Today, we choose the triple lake hike of Nymph, Dream, and Emerald Lakes.
Nymph Lake - the smallest of the triple lakes it is covered by lily pads in summer and in July these bear a beautiful the yellow flower.
Dream Lake - the largest of the lakes on this hike, it is a favorite spot for fishermen. A few years back I came to Rocky in the company of two fly fishermen (I wrote while they fished, 12 hours whew!). Today when Barbara and I arrive there is a conclave of Hebrew students (forelocks, yarmulkes, and black derby hats). Their unexpected presence in this western setting was underlined by the fact that most of them had wonderful New York accents.
Emerald Lake - every time I visit this last gem on the triple string I'm always tempted to think 'this is a place where elves would come to dance.' On the trail to this final lake we pass a waterfall and a wetland meadow alive with butterflies. At the lake we meet folks from all over the country, including a married couple from North Liberty Iowa. We come to find out they often frequented a restaurant owned by my wife's brother.
Saturday, July 12, 2014
Book review: Beautiful Geometry by Eli Maor and Eugen Jost.
For those of you have followed any of my math problem posts on Facebook, you know that I am a sucker for the art of Mathematics. I love patterns, whether they be geometric, fractal (the Snowflake Curve or Sierpinski's Triangle), number theory, number images (see the string art representation of Pi on my Facebook home page), or just plain weird math factoids (like those surrounding prime numbers), I am still, at 63 years of age, delighted by the artistry of math.
Which is why I am doubly delighted by Maor and Jost's wonderful new book, 'Beautiful Geometry'. First of all, pretty much all of the examples I gave in the first paragraph can be found in the 51 vignettes expounded upon in the pages of this treasure. Each of these mini-expositions explores a fascinating bit of math lore. One such vignette surrounded a fact that drew me to Geometry long ago when I was in high school.
If you will allow me I will attempt to delineate it then explain it, hopefully, dear reader, to your satisfaction. Here goes:
DRAW A QUADRILATERAL (A FIGURE WITH 4 SIDES). FIND THE MIDPOINT OF EACH SIDE AND CONNECT THEM IN CONSECUTIVE ORDER. THE RESULTING FIGURE THAT IS INSCRIBED (IN THE INTERIOR OF THE QUADRILATERAL) IS ALWAYS A PARALLELOGRAM (4 SIDES, OPPOSITE SIDES PARALLEL).
On first blush one is tempted to choose ordinary quadrilaterals: squares, rectangles, other parallelograms, and it is easy to see that the theorem holds up. But then I would challenge you to move on to more exotic Quads: Trapezoids, Quads with no sides congruent, even non-convex Quads such as chevrons (think the shirt front communicators on Star Trek). The theorem is true in these cases as well.
When I first saw this back in my salad days, I was entranced. How could this be? How could these seemingly unrelated shapes (OK they are related; they're all 4-sided) all produce this wonderful result. From that moment on I was hooked. I stepped foot onto a path of discovery that stretched back to antiquity and into the minds of some of the greatest thinkers ever to draw the same oxygen into their lungs that I did. In 63 years on the planet I have not been disappointed.
Oh, in case you think I forgot about Jost and Maor, the piece I just described was just one of the sweet little gems in 'Beautiful Geometry' - in fact it is the 3rd vignette.
The book is enhanced by splendidly colorful graphics exemplifying the patterns and results of thousands of years of mathematics. In the first short piece of 'Beautiful Geometry' we are introduced to one of the founding fathers of Geometry, Thales of Miletus. Before you even step foot into the bit of mathematical whimsy the authors bring to light, you are introduced to this man who lived two and a half thousand years ago. The authors tell us that Thales wasn't the first person to do high level Mathematics - the Egyptians did amazing math, think pyramids - but he might have been the first to speak of the artistry and beauty of math outside of it's practical uses. He asked why something was true and and attempted to prove it from fundamental principals.
I'll leave you with the center piece of this vignette on Thales (which by the way has intrigued me for all my days).
DRAW A CIRCLE AND DRAW IN IT'S DIAMETER (THE LINE SEGMENT THAT CUTS THE CIRCLE IN HALF). IF YOU CHOOSE ANY POINT ON THE CIRCLE AND CONNECT IT TO THE TWO END POINTS OF THE DIAMETER YOU WILL ALWAYS GET A RIGHT TRIANGLE.
When, long ago, when I was introduced to this result I had to test it for myself. I picked points straight across from the circle's center and points relatively near the diameter itself. The results were always the same, a right triangle. How cool is that? But Jost and Maor up the cool factor several increments by some wonderful graphics showing not only Thales' results but also the results of mathematicians who built on the works of this long ago geometer.
Well, I promised you I would end with the beginning of 'Beautiful Geometry' and I will. Take a look at this marvelous book. Regardless of how you feel about math, you will find something within it pages to bring a smile to your face.
Which is why I am doubly delighted by Maor and Jost's wonderful new book, 'Beautiful Geometry'. First of all, pretty much all of the examples I gave in the first paragraph can be found in the 51 vignettes expounded upon in the pages of this treasure. Each of these mini-expositions explores a fascinating bit of math lore. One such vignette surrounded a fact that drew me to Geometry long ago when I was in high school.
If you will allow me I will attempt to delineate it then explain it, hopefully, dear reader, to your satisfaction. Here goes:
DRAW A QUADRILATERAL (A FIGURE WITH 4 SIDES). FIND THE MIDPOINT OF EACH SIDE AND CONNECT THEM IN CONSECUTIVE ORDER. THE RESULTING FIGURE THAT IS INSCRIBED (IN THE INTERIOR OF THE QUADRILATERAL) IS ALWAYS A PARALLELOGRAM (4 SIDES, OPPOSITE SIDES PARALLEL).
On first blush one is tempted to choose ordinary quadrilaterals: squares, rectangles, other parallelograms, and it is easy to see that the theorem holds up. But then I would challenge you to move on to more exotic Quads: Trapezoids, Quads with no sides congruent, even non-convex Quads such as chevrons (think the shirt front communicators on Star Trek). The theorem is true in these cases as well.
When I first saw this back in my salad days, I was entranced. How could this be? How could these seemingly unrelated shapes (OK they are related; they're all 4-sided) all produce this wonderful result. From that moment on I was hooked. I stepped foot onto a path of discovery that stretched back to antiquity and into the minds of some of the greatest thinkers ever to draw the same oxygen into their lungs that I did. In 63 years on the planet I have not been disappointed.
Oh, in case you think I forgot about Jost and Maor, the piece I just described was just one of the sweet little gems in 'Beautiful Geometry' - in fact it is the 3rd vignette.
The book is enhanced by splendidly colorful graphics exemplifying the patterns and results of thousands of years of mathematics. In the first short piece of 'Beautiful Geometry' we are introduced to one of the founding fathers of Geometry, Thales of Miletus. Before you even step foot into the bit of mathematical whimsy the authors bring to light, you are introduced to this man who lived two and a half thousand years ago. The authors tell us that Thales wasn't the first person to do high level Mathematics - the Egyptians did amazing math, think pyramids - but he might have been the first to speak of the artistry and beauty of math outside of it's practical uses. He asked why something was true and and attempted to prove it from fundamental principals.
I'll leave you with the center piece of this vignette on Thales (which by the way has intrigued me for all my days).
DRAW A CIRCLE AND DRAW IN IT'S DIAMETER (THE LINE SEGMENT THAT CUTS THE CIRCLE IN HALF). IF YOU CHOOSE ANY POINT ON THE CIRCLE AND CONNECT IT TO THE TWO END POINTS OF THE DIAMETER YOU WILL ALWAYS GET A RIGHT TRIANGLE.
When, long ago, when I was introduced to this result I had to test it for myself. I picked points straight across from the circle's center and points relatively near the diameter itself. The results were always the same, a right triangle. How cool is that? But Jost and Maor up the cool factor several increments by some wonderful graphics showing not only Thales' results but also the results of mathematicians who built on the works of this long ago geometer.
Well, I promised you I would end with the beginning of 'Beautiful Geometry' and I will. Take a look at this marvelous book. Regardless of how you feel about math, you will find something within it pages to bring a smile to your face.
Saturday, October 26, 2013
Peanut Butter Sandwiches
Just the facts ma'am:
900 total Peanut Butter sandwiches of varying ilks
600 Peanut Butter and Jelly - PBJ
390 Peanut Butter and Banana - PBB
400 Peanut Butter and Pickle - PBP
250 Peanut Butter, Jelly, and Banana - PBJB
150 Peanut Butter, Jelly, and Pickle - PBJP
200 Peanut Butter, Banana, and Pickle - PBBP
20 Peanut Butter, Jelly, Banana, and Pickle - PBJBP
Remember that any grouping is included inside a larger group. Such as the 250 Peanut Butter, Jelly, and Banana, is by definition Peanut Butter and Jelly so included in that group of 600. Think in terms of a Venn diagram.
Now the question is: How many PB sandwiches are neither Jelly, Banana, or Pickle?
And we're off:
Start with the center of our three circle Venn diagram - 20 PBJBP, which means there are 180 PBBP, 130 PBJP, and 230 PBJB because each of these Venn intersections contain the 20 PBJBP.
Moving outward into the larger circle sections of our Venn Diagram we get 170 PBJ, 70 PBP, and 10 PBB. Now to compute the remaining Peanut Butter sandwiches we subtract all these sections from our initial 900 sandwiches.
900- 20 - 180 - 130 - 230 - 10 - 70 - 170 = 90 remaining PB sandwiches.
Now the question is: How many PB sandwiches are neither Jelly, Banana, or Pickle?
And we're off:
Start with the center of our three circle Venn diagram - 20 PBJBP, which means there are 180 PBBP, 130 PBJP, and 230 PBJB because each of these Venn intersections contain the 20 PBJBP.
Moving outward into the larger circle sections of our Venn Diagram we get 170 PBJ, 70 PBP, and 10 PBB. Now to compute the remaining Peanut Butter sandwiches we subtract all these sections from our initial 900 sandwiches.
900- 20 - 180 - 130 - 230 - 10 - 70 - 170 = 90 remaining PB sandwiches.
Subscribe to:
Posts (Atom)